Periodic cyclic homology of affine Hecke algebras
PPeriodic cyclic homology of affine Hecke algebras
Maarten Solleveld a r X i v : . [ m a t h . R T ] O c t eriodic cyclic homology of affine Hecke algebras / Maarten Solleveld, 2007 -253 p. : fig. ; 24 cm. - Proefschrift Universiteit van Amsterdam -Met samenvatting in het Nederlands.ISBN 978-90-9021543-3Cover designed with courtesy of William Wenger eriodic cyclic homologyofaffine Hecke algebras ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctoraan de Universiteit van Amsterdamop gezag van de Rector Magnificusprof. dr. J.W. Zwemmerten overstaan van een door hetcollege voor promoties ingestelde commissie,in het openbaar te verdedigen in de Aula der Universiteitop dinsdag 6 maart 2007, te 12:00 uurdoor
Maarten Sander Solleveld geboren te Amsterdam romotiecommissie:
Promotor: Prof. dr. E.M. OpdamCo-promotor: Prof. dr. N.P. LandsmanOverige leden: Prof. dr. G.J. HeckmanProf. dr. T.H. KoornwinderProf. dr. R. MeyerProf. dr. V. NistorDr. M. CrainicDr. H.G.J. PijlsDr. J.V. StokmanFaculteit der Natuurwetenschappen, Wiskunde en InformaticaDit proefschrift werd mede mogelijk gemaakt door de Nederlandse Organisatie voorWetenschappelijk Onderzoek (NWO). Het onderzoek vond plaats in het kader vanhet NWO Pionier project ”Symmetry in Mathematics and Mathematical Physics”. reface
The book you’ve just opened is the result of four years of research at the Korteweg-de Vries Institute for Mathematics of the Universiteit van Amsterdam. With thisthesis I hope to obtain the degree of doctor.Due to the highly specialized nature of my research, quite some mathematicalbackground is required to read this book. If you want to get an idea of what it’sabout, then you may find it useful to start with the Dutch summary.I would like to use this opportunity to express my gratitude to all those whosupported me in this work, intentionally or not.First of course my advisor, Eric Opdam, without whom almost my entire re-search would have been impossible. Over the years we had many pleasant con-versations, not only about mathematical issues, but also about chess, movies,education, Japan, and many other things. The intensity of our contact fluctuateda lot. Sometimes we didn’t talk for weeks, while in other periods we spoke manyhours, discussing our new findings every day.Vividly I recall one particular evening. After a prolonged discussion I had fi-nally returned home and was preparing dinner. Then unexpectedly Eric called totell me about some further calculations. Watching the boiling rice with one eyeand trying to visualize an affine Hecke algebra with the other, I quickly decidedthat I had to opt for dinner this time. Nevertheless that phonecall had a profoundinfluence on Section 6.6. I admire Eric’s deep mathematical insight, with whichhe managed to put me on the right track quite a few times.Also I would like to thank all the members of the promotion committee for thetime and effort they made to read the manuscript carefully. Niels Kowalzig was sokind to read and comment on the second chapter. For the summary I am indebtedto Klaas Slooten. His thesis was a source of inspiration, even though I didn’t usemany theorems from it.The KdVI is such a nice place that I come there for more than nine yearsalready. Everybody who worked there is responsible for that, but in particularErdal, Fokko, Geertje, Harmen, Misja, Peter, Rogier and Simon.Both my ushers, Mariska Berthol´ee-de Mie and Ionica Smeets, are very dear5o me. For being such good friends, for having faith in me, for lending me an ear,for cheering me up when things did not go as I would have liked. I am very happythat they will support me during the defence of this thesis.Furthermore I thank all my friends at US badminton for lots of (sporting)pleasure. Especially Paul den Hertog, who also took care of printing this book.To Karel van der Weide I am grateful for sharing his laconic yet hilarious viewson the chess world, on internetdating and on life in general.Bill Wenger was very generous in granting me permission to use his artworkon the cover of this book.But above all I thank Lieske Tibbe, for being my mother, and everything thatnaturally comes with that. If there is anybody who gave me the right scientificattitude to complete this thesis, it’s her.Amsterdam, January 2007 ontents
Preface 51 Introduction 92 K -theory and cyclic type homology theories 13 K -theory and the Chern character . . . . . . . . . . . 352.5 Equivariant cohomology and algebras of invariants . . . . . . . . . 45 p -adic groups 103 K -theoretic conjectures . . . . . . . . . . . . . . . . . . . . . . . . 160 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.2 GL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1796.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.4 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1917 6.5 GL n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056.6 A n − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2096.7 B n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 A Crossed products 223Bibliography 227Index 237Samenvatting 245Curriculum vitae 253 hapter 1
Introduction
This thesis is about a very interesting kind of algebras, Hecke algebras. They ap-pear in various fields of mathematics, for example knot theory, harmonic analysis,special functions and noncommutative geometry. The motivation for the researchpresented here lies mainly in the harmonic analysis of reductive p -adic groups.The description and classification of smooth representations of such groups is along standing problem. This is motivated by number theoretic investigations.The category of smooth representations is divided in certain blocks called Bern-stein components. It is known that in many cases Bernstein components can bedescribed with the representation theory of certain affine Hecke algebras. Thistranslation is a step forward, since in a sense affine Hecke algebras are muchsmaller and easier to handle than reductive p -adic groups.Hence it is desirable to obtain a good description of all irreducible represen-tations of an affine Hecke algebra. Such an algebra H ( W, q ) can be considered asa deformation of the (complex) group algebra of an affine Weyl group W , whichinvolves a few parameters q i ∈ C × . Let us briefly mention what is known aboutthe classification of its spectrum in various cases. All parameters q i equal to 1.In this special case H ( W, q ) is just the group algebra of W . The representa-tions of groups like W have been known explicitly for a long time, alreadyfrom the work of Clifford [31]. All parameters q i equal to the same complex number, not a root of unity.With the use of equivariant K -theory, Kazhdan and Lusztig [76] gave acomplete classification of the irreducible representations of H ( W, q ). It turnsout that they are in bijection with the irreducible representations of W . Thisbijection can be made explicit with Lusztig’s asymptotic Hecke algebra [84]. Exceptional cases.These may occur for example if there exist integers n i such that (cid:81) i q n i / i (cid:54) = 1is a root of unity, cf. page 132. The affine Hecke algebras for such parameters90 Chapter 1. Introductionmay differ significantly from those in the other cases, so we will not studythem here. General positive parameters.Quite strong partial classifications are available, mainly from the work ofDelorme and Opdam [39, 40]. General unequal parameters.These algebras have been studied in particular by Lusztig [88]. RecentlyKato [74] parametrized the representations of certain affine Hecke algebraswith three independent parameters, extending ideas that were used in case2).The affine Hecke algebras that arise from reductive p -adic groups have ratherspecial parameters: they are all powers of the cardinality of the residue field of the p -adic field. Sometimes they are all equal, and sometimes they are not, so thesealgebras are in case 4).Lusztig [88, Chapter 14] conjectured that Conjecture 1.1
For general unequal parameters there also exists an asymptoticHecke algebra. It yields a natural bijection between the irreducible representationsof H ( W, q ) and those of W . We are mainly concerned with a somewhat weaker version:
Conjecture 1.2
For positive parameters there is an isomorphism between theGrothendieck groups of finite dimensional H ( W, q ) -modules and of finite dimen-sional W -modules. In principle the verification of this conjecture would involve two steps a) Assign a W -representation to each (irreducible) H ( W, q )-representation, insome natural way (or the other way round). b) Prove that this induces an isomorphism on the above Grothendieck groups.In our study we make use of a technique that is obviously not available for p -adic groups, we deform the parameters continuously. We would like to do this inthe context of topological algebras, preferably operator algebras. For this reason,and to avoid the exceptional cases 3), we assume throughout most of the book thatall q i are positive real numbers. It was shown in [98] that for such parameters thereis a nice natural way to complete H ( W, q ) to a Schwartz algebra S ( W, q ) (thesenotations are preliminary). We will compare this algebra to the Schwartz algebra S ( W ) of the group W . Using the explicit description of S ( W, q ) given by Delormeand Opdam [39], in Section 5.3 we construct a Fr´echet algebra homomorphism φ : S ( W ) → S ( W, q ) (1.1)1with good properties. This provides a map from S ( W, q )-representations to S ( W )-representations. Together with the Langlands classification [40, Section 6] forrepresentations of H ( W, q ) and of W this takes care of a).To reformulate b) in more manageable terms we turn to noncommutative ge-ometry. There are (at least) three functors which are suited to deal with suchproblems: periodic cyclic homology ( HP ), in either the algebraic or the topologi-cal sense, and topological K -theory. Conjecture 1.3
There are natural isomorphisms1. HP ∗ ( H ( W, q )) ∼ = HP ∗ ( C [ W ]) HP ∗ ( S ( W, q )) ∼ = HP ∗ ( S ( W )) K ∗ ( S ( W, q )) ∼ = K ∗ ( S ( W ))The relation with Conjecture 1.2 is as follows. Although HP ∗ and K ∗ are functorson general noncommutative algebras, in our setting they depend essentially only onthe spectra of the algebras that we are interested in. These spectra are ill-behavedas topological spaces: the spectrum (cid:92) H ( W, q ) is a non-separated algebraic variety,while (cid:92) S ( W, q ) is a compact non-Hausdorff space. Nevertheless topological K -theory and periodic cyclic homology can be considered as cohomology theories onsuch spaces. With this interpretation Conjecture 1.3 asserts that the ”cohomologygroups” of (cid:92) H ( W, q ) and of (cid:92) S ( W, q ) are invariant under the deformations in theparameters q i . Contrarily to what one would expect from the results on page 9,from this noncommutative geometric point of view the algebras S ( W, q ) actuallybecome easier to understand when the parameters q i have less relations amongthemselves.For equal parameters Conjecture 1.3 has been around for a while. Part 3already appeared in the important paper [5], while part 1 was proven by Baum andNistor [8]. The proof relies on the aforementioned results of Kazhdan and Lusztig.In the unequal parameter case Conjecture 1.3.3 was formulated independently byOpdam [98, Section 1.0.1].In this thesis we make the following progress concerning these conjectures. InSection 3.4 we prove that there are natural isomorphisms HP ∗ ( H ( W, q )) ∼ = HP ∗ ( S ( W, q )) ∼ = K ∗ ( S ( W, q )) ⊗ Z C (1.2)Hence parts 1 and 2 of Conjecture 1.3 are equivalent, and both are weaker thanpart 3. Moreover in Section 5.4 we show that the Conjectures 1.2 and 1.3.3 areequivalent.Guided by these considerations we propose the following refined conjecture,which has also been presented by Opdam [99, Section 7.3]: Conjecture 1.4
The natural map K ∗ ( φ ) : K ∗ ( S ( W )) → K ∗ ( S ( W, q )) is an isomorphism for all positive parameters q i . K -theory and periodic cyclic homology can beconsidered as cohomology theories for certain non-Hausdorff spaces. Among otherswe prove comparison theorems like (1.2) for more general classes of algebras, allderived from so-called finite type algebras [77].In the next chapter we introduce affine Hecke algebras. A large part of the ma-terial presented here relies on the work of Opdam, in collaboration with Delorme,Heckman, Reeder and Slooten. We study the representation theory of affine Heckealgebras, which provides a clear picture of their spectra as topological spaces. Weare especially interested in the image of S ( W, q ) under the Fourier transform, asthis turns out to be an algebra of the type that we studied in Chapter 2. Weconclude with the above isomorphisms (1.2).These are also interesting because they can be generalized to algebras asso-ciated with reductive p -adic groups, which we will do in Chapter 4. Given areductive p -adic group G we recall the constructions of its Hecke algebra H ( G ), itsSchwartz algebra S ( G ) and its reduced C ∗ -algebra C ∗ r ( G ). The main new resultsin this chapter are natural isomorphisms HP ∗ ( H ( G )) ∼ = HP ∗ ( S ( G )) ∼ = K ∗ ( C ∗ r ( G )) ⊗ Z C (1.3)These have some consequences in relation with the Baum-Connes conjecture for G . In Chapter 5 we really delve into the study of deformations of affine Heckealgebras. The Fr´echet space underlying S ( W, q ) is independent of q , and we showthat all the (topological) algebra operations in S ( W, q ) depend continously on q .After that we focus on parameter deformations of the form q → q (cid:15) with (cid:15) ∈ [0 , (cid:15) we construct isomorphisms φ (cid:15) : S ( W, q (cid:15) ) → S ( W, q ) (1.4)that depend continuously on (cid:15) . The limit lim e ↓ φ (cid:15) is well-defined and indeed is(1.1). Furthermore we elaborate on the conjectures mentioned in this introduction.In support of the these conjectures, and to show what the techniques we de-veloped are up to, we dedicate Chapter 6 to calculations for affine Hecke algebrasof classical type. We verify Conjecture 1.4 in some low-dimensional cases and fortypes GL n and A n .We conclude the purely scientific part of the book with a short appendix. Itcontains some rather elementary results on crossed product algebras that are usedat various places. hapter 2 K -theory and cyclic typehomology theories This chapter is of a more general nature than the rest of this book. We startwith the study of some important covariant functors on the category of complexalgebras. These are Hochschild homology, cyclic homology and periodic cyclichomology. Contravariant versions of these functors also exist, but we will leavethese aside. All these functors go together by the name of cyclic theory.It is well-known that cyclic homology is related to K -theory by a naturaltransformation of functors called the Chern character. We are not satisfied with K -theory for Banach algebras, but instead study its extension to the larger categoriesof Fr´echet algebras or even m-algebras. From these abstract considerations we willsee that there are three functors which share almost identical properties: a) periodic cyclic homology, purely algebraically b) periodic cyclic homology, with the completed projective tensor product c) K -theory for Fr´echet algebrasThese functors can be regarded as noncommutative analogues of De Rham cohomology in the algebraic sense, for complex affine varieties De Rham cohomology in the differential geometric sense, for smooth manifolds K -theory for topological spacesBy a comparison theorem of Deligne and Grothendieck 1) and 2) agree for acomplex affine variety. For smooth manifolds 2) and 3) (with real coefficients)give the same result, essentially because both are generalized cohomology theories.This is also the reason that both can be computed as ˇCech cohomology of a constant sheaf134 Chapter 2. K -theory and cyclic type homology theoriesA noncommutative analogue of 4) does not appear to exist, so we develop it. Itwill be a sheaf that depends only on the spectrum of an algebra. Then we canalso consider d) ˇCech cohomology of this sheafThe main goal of this chapter is to generalize the isomorphisms between 1) - 4)to the setting of noncommutative algebras. So far this has been done only for b)and c).Let us also give a more concrete overview of this chapter. We start by recallingthe definitions and properties of cyclic theory in the purely algebraic sense. Thenwe specialize to finite type algebras, mainly following [77]. For such algebras wedefine a sheaf which provides the isomorphism between a) and d).After that we move on to topological algebras, especially Fr´echet algebras.Most of the properties of algebraic cyclic theory have been carried over to thistopological setting, but unfortunately these results have been scattered throughoutthe literature. We hope that bringing them together will serve the reader. We alsorecall several results concerning K -theory for Fr´echet algebras, which are mostlydue to Cuntz [33] and Phillips [102].In the final section we have to decide for what kind of topological algebras wewant to compare b), c) and d). Natural candidates are algebras that are finitelygenerated as modules over their center. For finitely generated (non-topological)algebras this condition leads to the aforementioned finite type algebras. Theirspectrum has the structure of a non-separated complex affine variety.In the topological setting we need to impose more conditions. Cyclic homologyworks best if there is a kind of smooth structure, so our topological analogue of afinite type algebra is of the form C ∞ ( X ; M N ( C )) G (2.1)where X is a smooth manifold and G a finite group. The action of G is a combi-nation of an action on X and conjugation by certain matrices.The comparisons between 1) - 4) all rely on triangulations and Mayer-Vietorissequences. We will apply these techniques to X in a suitable way. This will enableus to define d) and prove that it gives the same results as b) and c). Moreoverwe prove that if X happens to be a complex affine variety, then there is a naturalisomorphism HP ∗ (cid:0) O ( X ; M N ( C )) G (cid:1) ∼ −−→ HP ∗ (cid:0) C ∞ ( X ; M N ( C )) G (cid:1) (2.2)The O stands for algebraic functions, so the left hand side corresponds to a) and1) above..1. Algebraic cyclic theory 15 We give the definitions and most important properties of Hochschild homology,cyclic homology and periodic cyclic homology. We do this only in the category ofalgebras over C , although quite a big deal of the theory is also valid over arbitraryfields. We will be rather concise, referring to Loday’s monograph [81] for morebackground and proofs.Cyclic homology was discovered more or less independently by several people,confer the work of Connes [32], Loday and Quillen [82] and Tsygan [129]. We willdefine it with the so-called cyclic bicomplex. So let n ∈ N , let A be any complexalgebra and let A ⊗ n be the n -fold tensor product of A over C . We abbreviatean elementary tensor a ⊗ · · · ⊗ a n to ( a , . . . , a n ) and we define linear operators b, b (cid:48) : A ⊗ n +1 → A ⊗ n and λ, N : A ⊗ n → A ⊗ n by the following formulas: b (cid:48) ( a , a , . . . , a n ) = n − (cid:88) i =0 ( − i ( a , . . . , a i − , a i a i +1 , a i +2 , . . . , a n ) b ( a , a , . . . , a n ) = b (cid:48) ( a , a , . . . , a n ) + ( − n ( a n a , a , . . . , a n − ) λ ( a , . . . , a n ) = ( − n − ( a n , a , . . . , a n − ) N = 1 + λ + · · · + λ n − (2.3)For unital A we also define s, B : A ⊗ n → A ⊗ n +1 : s ( a , . . . , a n ) = (1 , a , . . . , a n ) B = (1 − λ ) sN (2.4)These are the ingredients of a bicomplex CC per ( A ) : · · · · · · · · · · · ·↓ ↓ ↓ ↓ · · · ← A ⊗ N ←− A ⊗ − λ ←−−− A ⊗ N ←− A ⊗ ← · · ·↓ − b (cid:48) ↓ b ↓ − b (cid:48) ↓ b · · · ← A ⊗ N ←− A ⊗ − λ ←−−− A ⊗ N ←− A ⊗ ← · · ·↓ − b (cid:48) ↓ b ↓ − b (cid:48) ↓ b · · · ← A N ←− A − λ ←−−− A N ←− A ← · · · − (2.5)The indicated grading means that CC perp,q ( A ) = A ⊗ q +1 .Consider also the subcomplexes CC ( A ), consisting of all the columns labelledby p ≥
0, and CC { } ( A ), which consists only of the columns numbered 0 and 1.6 Chapter 2. K -theory and cyclic type homology theoriesWith these bicomplexes we associate differential complexes with a single grading.Their spaces in degree n are CC { } n ( A ) = CC per ,n ( A ) ⊕ CC per ,n − ( A ) = A ⊗ n +1 ⊕ A ⊗ n CC n ( A ) = n (cid:77) p =0 CC perp,n − p ( A ) = A ⊗ n +1 ⊕ A ⊗ n ⊕ · · · ⊕ ACC pern ( A ) = (cid:89) p + q = n CC perp,q ( A ) = (cid:89) q ≥ A ⊗ q +1 (2.6)This enables us to define the Hochschild homology HH n ( A ), the cyclic homology HC n ( A ) and the periodic cyclic homology HP n ( A ) : HH n ( A ) = H n ( CC { }∗ ( A )) HC n ( A ) = H n ( CC ∗ ( A )) HP n ( A ) = H n ( CC per ∗ ( A )) (2.7)Since all the above complexes are functorial in A these homology theories areindeed covariant functors. The definitions we gave are neither the simplest possibleones, nor the best for explicit computations, but they do have the advantage thatthey work for every algebra, unital or not.By the way, we can always form the unitization A + . This is the vector space C ⊕ A with multiplication( z , a )( z , a ) = ( z z , z a + z a + a a ) (2.8)Clearly every algebra morphism φ : A → B gives a unital algebra morphism φ + : A + → B + . There are natural isomorphisms HH n ( A ) ∼ = coker (cid:0) HH n ( C ) → HH n ( A + ) (cid:1) ∼ = ker (cid:0) HH n ( A + ) → HH n ( C ) (cid:1) (2.9)and similarly for HC n and HP n .Often we shall want to consider all degrees at the same time, and for thispurpose we write HH ∗ ( A ) = (cid:77) n ≥ HH n ( A ) HC ∗ ( A ) = (cid:77) n ≥ HC n ( A )The map S : CC perp,q ( A ) → CC perp − ,q ( A ) simply shifting everything two columns tothe left is clearly an automorphism of CC per ( A ). Moreover it decreases the degreeby two, so it induces a natural isomorphism HP n ( A ) ∼ −−→ HP n − ( A ) (2.10).1. Algebraic cyclic theory 17Thus we may consider periodic cyclic homology as a Z / Z -graded functor, or wemay restrict n to { , } . In particular we shall write HP ∗ ( A ) = HP ( A ) ⊕ HP ( A )Regarding CC ( A ) as a quotient of CC per ( A ), we get an induced map ¯ S : CC ( A ) → CC ( A ). This map is surjective, and its kernel is exactly CC { } ( A ). This leads toConnes’ periodicity exact sequence : · · · → HH n ( A ) I −→ HC n ( A ) S −→ HC n − ( A ) B −→ HH n − ( A ) → · · · (2.11)Here I comes from the inclusion of CC { } ( A ) in CC ( A ) and B is induced by themap from (2.4). In combination with the five lemma this is a very useful tool; itenables one to prove easily that many functorial properties of Hochschild homologyalso hold for cyclic homology.Furthermore we notice that the bicomplex CC per ( A ) is the inverse limit ofits subcomplexes S r ( CC ( A )). In many cases this gives an isomorphism between HP n ( A ) and lim ←− HC n +2 r ( A ). In general however it only leads to a short exactsequence0 → lim ∞← r HC n +1+2 r ( A ) → HP n ( A ) → lim ∞← r HC n +2 r ( A ) → ←− is the first derived functor of lim ←− , see [81, Propostion 5.1.9].Next we state some well-known features of the functors under consideration.1. Additivity. If A m ( m ∈ N ) are algebras then HH n (cid:32) ∞ (cid:77) m =1 A m (cid:33) ∼ = ∞ (cid:77) m =1 HH n ( A m ) HH n (cid:32) ∞ (cid:89) m =1 A m (cid:33) ∼ = ∞ (cid:89) m =1 HH n ( A m )and similarly for HC n and (cid:0) HP n , (cid:81) (cid:1) .2. Stability. If A is H-unital then HH n ( M m ( A )) ∼ = HH n ( A )More generally, if B and C are unital and Morita-equivalent, then HH n ( B ) ∼ = HH n ( C )These statements hold also for HC n and HP n .3. Continuity. If A = lim m →∞ A m is an inductive limit then HH n ( A ) ∼ = lim m →∞ HH n ( A m ) HC n ( A ) ∼ = lim m →∞ HC n ( A m )8 Chapter 2. K -theory and cyclic type homology theoriesHowever, HP ∗ is not continuous in general. A sufficient condition for con-tinuity can be found in [16, Theorem 3] : there exists a N ∈ N such that HH n ( A m ) = 0 ∀ n > N ∀ m .To an extension of algebras0 → A → B → C → → A → B + → C + → B and C are unital. It was discovered by Wodzicki that what we need for A is not somuch unitality, but a weaker notion called homological unitality, or H-unitalityfor short. It is easily seen that for unital algebras the map s defines a contractinghomotopy for the complex ( A ⊗ n , b (cid:48) ), and in fact with some slight modificationsthis construction also applies to algebras that have left or right local units. Thus,we call a complex algebra A H-unital if the homology of the complex ( A ⊗ n , b (cid:48) ) is0. Now Wodzicki’s excision theorem [136] says Theorem 2.1
Let → A → B → C → be an extension of algebras, with A H-unital. There exist long exact sequences · · · → HH n ( A ) → HH n ( B ) → HH n ( C ) → HH n − ( A ) → · · ·· · · → HC n ( A ) → HC n ( B ) → HC n ( C ) → HC n − ( A ) → · · ·· · · → HP n ( A ) → HP n ( B ) → HP n ( C ) → HP n − ( A ) → · · · It turns out [36] that for HP ∗ it is not necessary to require H-unitality. Dueto the 2-periodicity of this functor we get, for any extension of algebras, an exacthexagon HP ( A ) → HP ( B ) → HP ( C ) ↑ ↓ HP ( C ) ← HP ( B ) ← HP ( A ) (2.13)It will be very useful to combine the excision property with the five lemma: Lemma 2.2
Suppose we have a commutative diagram of abelian groups, with ex-act rows: A → A → A → A → A ↓ f ↓ f ↓ f ↓ f ↓ f B → B → B → B → B If f is surjective, f and f are isomorphisms and f is injective, then f is anisomorphism. .1. Algebraic cyclic theory 19Because we intend to apply the next result to several different functors, weformulate it very abstractly. Lemma 2.3
Let A and B be categories of algebras, and AG the category of abeliangroups. Suppose that F ∗ : A → AG and G ∗ : B → AG are Z -graded, covariantfunctors satisfying excision, and that T ∗ : F ∗ → G ∗ is a natural transformation ofsuch functors. Consider two sequences of ideals I ⊂ I ⊂ · · · ⊂ I n ⊂ I n +1 = A J ⊂ J ⊂ · · · ⊂ J n ⊂ J n +1 = B (2.14) in A and B respectively. If we have an algebra homomorphism φ : A → B suchthat φ ( I m ) ⊂ J m and T ( J m /J m +1 ) F ( φ ) = G ( φ ) T ( I m /I m +1 ) : F ( I m /I m +1 ) → G ( J m /J m +1 ) is an isomorphism ∀ m ≤ n , then T ( φ ) := T ( B ) F ( φ ) = G ( φ ) T ( A ) : F ( A ) → G ( B ) is an isomorphism. Similarly, consider two exact sequences → A → A → · · · → A n → → B → B → · · · → B n → in A and B . Suppose that we have a morphism of exact sequences ψ = ( ψ m ) nm =1 ,such that T ( ψ m ) : F ( A m ) → G ( B m ) is an isomorphism for all but one m . Then it is an isomorphism for all m .Proof. Consider the short exact sequences0 → I m − → I m → I m /I m − → → J m − → J m → J m /J m − → → im ( A m − → A m ) → A m → im ( A m → A m +1 ) → → im ( B m − → B m ) → B m → im ( B m → B m +1 ) → m = 1, so with induction we reduce the entire lemma to thestatement for exact sequences, with m = 3. Now we consider only the case where T ( ψ m ) is an isomorphism for m = 1 and m = 3, since the other cases are verysimilar. For any k ∈ Z we see from the commutative diagram F k +1 ( A ) → F k ( A ) → F k ( A ) → F k ( A ) → F k − ( A ) ↓ ↓ ↓ ↓ ↓ G k +1 ( B ) → G k ( B ) → G k ( B ) → G k ( B ) → G k − ( B )and Lemma 2.2 that T k ( ψ ) is an isomorphism. (cid:50) K -theory and cyclic type homology theoriesHaving elaborated a little on the functorial properties of HH ∗ , HC ∗ and HP ∗ ,we will show now what they look like on some nice algebras. First we fix thenotations of some well-known objects from algebraic geometry.Assume for the rest of this section that A is a commutative, unital complexalgebra. The A -module of K¨ahler differentials Ω ( A ) is generated by the symbols da , subject to the following relations, for any a, b ∈ A, z ∈ C : d ( za ) = z dad ( a + b ) = da + dbd ( ab ) = a db + b da (2.16)The A -module of differential n -forms is the n -fold exterior product over A :Ω n ( A ) = (cid:94) nA Ω ( A ) (2.17)and, just to be sure, we decree that Ω ( A ) = A . The formal operator d defines adifferential Ω n ( A ) → Ω n +1 ( A ) by d ( a da ∧ · · · ∧ da n ) = da ∧ da ∧ · · · ∧ da n (2.18)The De Rham homology of A is defined as H DRn ( A ) = H n (Ω ∗ ( A ) , d ) (2.19)If A = O ( V ) is the ring of regular functions on an affine complex algebraic variety V , not necessarily irreducible, then we also writeΩ n ( V ) = Ω n ( A ) and H nDR ( V ) = H DRn ( A )One can check that the following formulas define natural maps: HH n ( A ) → Ω n ( A ) : ( a , a , . . . , a n ) → a da ∧ · · · ∧ da n Ω n ( A ) → HH n ( A ) : a da ∧ · · · ∧ da n → (cid:80) σ ∈ S n (cid:15) ( σ )( a , a σ (1) , . . . , a σ ( n ) )(2.20)The celebrated Hochschild-Kostant-Rosenberg theorem [60] says that these mapsare isomorphisms if A is a smooth algebra. Yet the author believes that a precisedefinition of smoothness would digress too much, so we only mention that a typicalexample is O ( V ) with V nonsingular, and that all the details can be found in [81,Appendix E]. Anyway, under (2.20) the differential d corresponds to the map B from (2.4) and therefore the Hochschild-Kostant-Rosenberg theorem also gives the(periodic) cyclic homology of smooth algebras: HH n ( A ) ∼ = Ω n ( A ) (2.21) HC n ( A ) ∼ = Ω n ( A ) /d Ω n − ( A ) ⊕ H DRn − ( A ) ⊕ H DRn − ( A ) ⊕ · · · (2.22) HP n ( A ) ∼ = (cid:89) m ∈ Z H DRn +2 m ( A ) (2.23).2. Periodic cyclic homology of finite type algebras 21 The theory of finite type algebras was built by Baum, Kazhdan, Nistor and Schnei-der [8, 77]. This turns out to be a pleasant playground for cyclic theory, culmi-nating roughly speaking in the statement “the periodic cyclic homology of a finitetype algebra is an invariant of its spectrum.” We discuss this result, and someof its background. We also add one new ingredient to support this point of view,namely a sheaf, depending only on the spectrum of A , whose ˇCech cohomology isisomorphic to HP ∗ ( A ).All this is made possible by several extra features that HP ∗ possesses, comparedto HH ∗ and HC ∗ . Recall that an extension of algebras0 → I → A → A/I → I is nilpotent, i.e. if I n = 0 for some n ∈ N .An algebraic homotopy between two algebra homomorphisms f, g : A → B is a collection φ t : A → B of morphisms, depending polynomially on t , suchthat φ = f and φ = g . This is equivalent to the existence of a morphism φ : A → B ⊗ C [ t ] such that f = ev ◦ φ and g = ev ◦ φ .Goodwillie [49, Corollary II.4.4 and Theorem II.5.1] established two closelyrelated features: Theorem 2.4
The functor HP ∗ is homotopy invariant and turns nilpotent exten-sions into isomorphisms. Thus, with the above notation, HP ∗ ( f ) = HP ∗ ( g ) HP ∗ ( I ) = 0 and HP ∗ ( A ) ∼ −−→ HP ∗ ( A/I ) is an isomorphism. Homotopy invariance can be regarded as a special case of the K¨unneth theorem,which holds for periodic cyclic homology under some mild conditions.
Theorem 2.5
Suppose that A is a unital algebra such that • the lim ←− -term in (2.12) vanishes, i.e. HP n ( A ) ∼ = lim ∞← r HC n +2 r ( A ) • HP ∗ ( A ) has finite dimensionThen the K¨unneth theorem holds for HP ∗ ( A ) . This means that for any unitalalgebra B there is a natural isomorphism of Z / Z -graded vector spaces HP ∗ ( A ) ⊗ HP ∗ ( B ) ∼ −−→ HP ∗ ( A ⊗ B )2 Chapter 2. K -theory and cyclic type homology theories Proof.
See [70, Theorem 3.10] and [44, Theorem 4.2]. (cid:50)
Reconsider the Hochschild-Kostant-Rosenberg theorem (2.23) for the periodiccyclic homology of the ring of regular functions on a nonsingular affine complexvariety. It gives an isomorphism of Z / Z -graded vector spaces HP ∗ ( O ( V )) ∼ = H ∗ DR ( V ) (2.25)Now let V an be the set V endowed with its natural analytic topology. By afamous theorem of Grothendieck and Deligne (cf. [52] and [57, Theorem IV.1.1])the algebraic De Rham cohomology of V is naturally isomorphic to the analyticDe Rham cohomology of V an : H ∗ DR ( V ) ∼ = H ∗ DR ( V an ; C ) (2.26)As is well-known, all classical cohomology theories agree on the category of smoothmanifolds, for instance H ∗ DR ( V an ; C ) ∼ = ˇ H ∗ ( V an ; C ) (2.27)the latter denoting ˇCech cohomology with coefficients in C . Because of the similarfunctorial properties, it is not surprising that the composite isomorphism of (2.25)- (2.27) holds in greater generality. This was confirmed in [77, Theorem 9] : Theorem 2.6
Let X be an affine complex variety, I ⊂ O ( X ) an ideal and Y ⊂ X the subvariety defined by I . It is neither assumed that X is nonsingular orirreducible, nor that I is prime. There is a natural isomorphism HP n ( I ) ∼ = ˇ H [ n ] ( X an , Y an ; C ) := (cid:89) m ∈ Z ˇ H n +2 m ( X an , Y an ; C )Recall that a primitive ideal in a complex algebra is the kernel of a (nonzero)irreducible representation of A . The primitive ideal spectrum Prim( A ) is the set ofall primitive ideals of A , and the Jacobson radical Jac( A ) is the intersection of allthese primitive ideals. Note that every nilpotent ideal is contained in Jac( A ). Weendow Prim( A ) with the Jacobson topology, which means that all closed subsetsare of the form S := { I ∈ Prim( A ) : I ⊃ S } (2.28)for some subset S of A . Denote by d I the dimension of an irreducible representationwith kernel I ∈ Prim( A ). If d I < ∞ ∀ I then Prim( A ) is a T -space, but in generalit is only a T -space.For commutative A the primitive ideals are precisely the maximal ideals, andPrim( A ) is an algebraic variety. In this case there also is a natural topology onthe set Prim( A ) that makes it into an analytic variety, see [116, Section 5].If φ : A → B is an algebra homomorphism and J ∈ Prim( B ) then φ − ( J ) is anideal, but it is not necessarily primitive. So Prim is not a functor, it only induces.2. Periodic cyclic homology of finite type algebras 23a map J → φ − ( J ) from Prim( B ) to the power set of Prim( A ). However, if forevery J ∈ Prim( B ) there exists exactly one I ∈ Prim( A ) containing φ − ( J ), then φ does induce a continuous map Prim( B ) → Prim( A ) and we call φ spectrumpreserving.Now we give the definition of a finite type algebra. Let k be a finitely generatedcommutative unital complex algebra, i.e. the ring of regular functions on someaffine complex variety. A k -algebra is a (nonunital) algebra A together with aunital morphism from k to Z ( M ( A )), the center of the multiplier algebra of A .An algebra B is of finite type if there exists a k such that B is k -algebra which isfinitely generated as a k -module. An algebra morphism φ : A → B is a morphismof finite type algebras if it is k -linear, for some k over which both A and B are offinite type.As announced, the most important theorem in this category says that HP ∗ isdetermined by Prim, see [8, Theorems 3 and 8]. Theorem 2.7
Let φ : A → B be a spectrum preserving morphism of finite typealgebras. Then Prim( B ) → Prim( A ) is a homeomorphism and HP ∗ ( φ ) : HP ∗ ( A ) → HP ∗ ( B ) is an isomorphism. More generally, we might have ideals like in (2.14), such that the induced maps I m /I m +1 → J m /J m +1 are all spectrum preserving, but φ : A → B is not. In thatcase φ is called weakly spectrum preserving. By Theorem 2.7 and Lemma 2.3 suchmaps also induce isomorphisms on periodic cyclic homology.To understand this better we zoom in on the spectrum, relying heavily on [77,Section 1]. Until further notice we assume that A is a unital finite type algebra.The central character mapΘ : Prim( A ) → Prim( Z ( A )) : I → I ∩ Z ( A ) (2.29)is a finite-to-one continuous surjection. For k, p ∈ N we writePrim k ( A ) = { I ∈ Prim( A ) : d I = k } Prim ≤ p ( A ) = (cid:83) pk =1 Prim k ( A ) (2.30)The sets Prim ≤ p ( A ) are all closed and, as the frequent occurence of the word“finite” already suggests, there exists a N A ∈ N such that Prim ≤ N A ( A ) = Prim( A ).This leads to the so-called standard filtration of A : A = I st ⊃ I st ⊃ · · · ⊃ I stN A − ⊃ I stN A = Jac( A ) I stp = (cid:92) d I ≤ p I = { a ∈ A : π ( a ) = 0 if π is a representation with dim π ≤ p } (2.31)Observe that Prim (cid:0) I stq /I stp (cid:1) = p (cid:91) k = q +1 Prim k ( A ) (2.32)4 Chapter 2. K -theory and cyclic type homology theoriesFrom (2.31) we also get a filtration of the cyclic bicomplex : CC per ∗ ( A ) = CC per ∗ ( A ) ⊃ CC per ∗ ( A ) ⊃ · · · ⊃ CC per ∗ ( A ) N A − ⊃ CC per ∗ ( A ) N A CC per ∗ ( A ) p = ker (cid:0) CC per ∗ ( A ) → CC per ∗ (cid:0) A/I stp (cid:1)(cid:1) (2.33)Using standard (but involved) techniques from homological algebra we con-struct a spectral sequence E p,qr with E p,q = CC per ∗ ( A ) p − /CC per ∗ ( A ) p ∼ = ker (cid:0) CC per − p − q (cid:0) A/I stp (cid:1) → CC per − p − q (cid:0) A/I stp − (cid:1)(cid:1) (2.34) E p,q = HP − p − q (cid:0) I stp − /I stp (cid:1) (2.35) E p,q ∞ = HP − p − q (cid:0) I stp − (cid:1) /HP − p − q (cid:0) I stp (cid:1) (2.36)Moreover d E : E p,q → E p,q +10 comes directly from the differential in the cyclicbicomplex and d E : E p,q → E p +1 ,q is the composition HP − p − q (cid:0) I stp − /I stp (cid:1) → HP − p − q (cid:0) A/I stp (cid:1) → HP − p − q − (cid:0) I stp /I stp +1 (cid:1) of the map induced by the inclusion I stp − /I stp → A/I stp and the connecting map ofthe extension 0 → I stp /I stp +1 → A/I stp +1 → A/I stp →
0A most pleasant property of the standard filtration (2.31) is that the quotients I stp − /I stp behave like commutative algebras. More precisely, consider the analyticspace X p associated to Prim (cid:0) Z (cid:0) A/I stp (cid:1)(cid:1) , and its subvariety Y p = (cid:8) I ∈ X p : Z (cid:0) A/I stp (cid:1) ∩ I p − /I p ⊂ I (cid:9) (2.37)The central character map for A/I stp defines a bijectionPrim p ( A ) = Prim (cid:0) I stp − /I stp (cid:1) → X p \ Y p (2.38)and according to [77, Theorem 1] there is a natural isomorphism E p,q = HP − p − q (cid:0) I stp − /I stp (cid:1) ∼ = ˇ H [ p + q ] ( X p , Y p ; C ) (2.39)Comparing this with Theorem 2.6 we see that I stp − /I stp is indeed “close to com-mutative” in the sense that its periodic cyclic homology can be computed as theˇCech cohomology of a constant sheaf over its spectrum.We seek to generalize this to “less commutative”, nonunital finite type algebras.Let X be the set Prim( k ) with the analytic topology, and V ( A ) the set Prim( A )with the coarsest topology that makes Θ : Prim( A ) → X continuous and is finerthan the Jacobson topology. This topology depends only on the fact that A is afinite type algebra, and not on the particular choice of k . So if A is unital we mayjust as well assume that k = Z ( A ) and X = X ..2. Periodic cyclic homology of finite type algebras 25We will construct a sheaf A over X whose stalk at x is the (finite dimensional)complex vector space with basis Θ − ( x ). By definition all continuous sections ofthis collection of stalks are constructed from local sections of Θ : V ( A ) → X . Moreprecisely, given an open Y ⊂ X we call a section s of (cid:81) x ∈ Y A ( x ) → Y continuousat y ∈ Y if there exist • a neighborhood U of y in Y • connected components C , . . . , C n of Θ − ( U ), not necessarily different • for every i a section s i of the quotient map from C i to its Hausdorffization C Hi • complex numbers z , . . . , z n such that ∀ x ∈ U s ( x ) = n (cid:88) i =1 z i (cid:0) s i ( C Hi ) ∩ Θ − ( x ) (cid:1) (2.40)For example if X (cid:48) is a closed subvariety of X and A = { f ∈ O ( X ) : f ( X (cid:48) ) = 0 } then A is the direct image of the constant sheaf (with stalk C ) on X \ X (cid:48) .Notice that A is functorial in A . If φ : A → B is a morphism of finite type k -algebras and V is a left A -module, then B ⊗ A V is a B -module. If we consideronly the semisimple forms of these modules, then we get a homomorphism Z [Prim( A )] → Z [Prim( B )]which extends naturally to a morphism A → B of sheaves over X .The motivation for this sheaf comes from topological K -theory: the local sec-tions s i are supposed to model “local” idempotents in A . The classes of thesethings should generate HP ∗ ( A ), leading to Theorem 2.8
There is an unnatural isomorphism of finite dimensional vectorspaces HP ∗ ( A ) ∼ = ˇ H ∗ ( X ; A ) Proof.
Assume first that A is unital. Let A p be the sheaf (over X ) constructedfrom A/I stp in the same way as we constructed A from A ; it has stalks A p ( x ) = C { Θ − ( x ) ∩ Prim ≤ p ( A ) } (2.41)Since Prim ≤ p ( A ) is closed in Prim( A ) there is a natural surjection A → A p , whichcomes down to forgetting all primitive ideals I with d I > p . Thus we get filtrationsof the (pre)sheaf A : A = I ⊃ I ⊃ · · · ⊃ I N A − ⊃ I N A = 0 I p = ker ( A → A p ) (2.42)6 Chapter 2. K -theory and cyclic type homology theoriesand of the ˇCech complex ˇ C ∗ ( X ; A ) (this is a pretty complicated object, see [47, § C ∗ ( X ; A ) = ˇ C ∗ ( X ; A ) ⊃ ˇ C ∗ ( X ; A ) ⊃ · · · ⊃ ˇ C ∗ ( X ; A ) N A − ⊃ ˇ C ∗ ( X ; A ) N A = 0ˇ C ∗ ( X ; A ) p = ker (cid:0) ˇ C ∗ ( X ; A ) → ˇ C ∗ ( X ; A p ) (cid:1) ∼ = ˇ C ∗ ( X ; I p ) (2.43)The presheaf B p := ker( A p → A p − ) is actually a sheaf, and it has stalks B p ( x ) = C { Θ − ( x ) ∩ Prim p ( A ) } (2.44)From these data we construct a spectral sequence F p,qr with terms F p,q = ˇ C p + q ( X ; A ) p − / ˇ C p + q ( X ; A ) p ∼ = ˇ C p + q ( X ; B p ) (2.45) F p,q = ˇ H p + q ( X ; B p ) (2.46) F p,q ∞ = ˇ H p + q ( X ; I p − ) / ˇ H p + q ( X ; I p ) (2.47)In this sequence d F : F p,q → F p,q +10 is the normal ˇCech differential, while d F : F p,q → F p +1 ,q is induced by the inclusion B p → A p and the connecting mapassociated to the short exact sequence0 → B p +1 → A p +1 → A p → B p we see thatthere are natural isomorphismsˇ C p + q ( X ; B p ) ∼ = ˇ C p + q ( X p , Y p ; C )ˇ H p + q ( X ; B p ) ∼ = ˇ H p + q ( X p , Y p ; C )Clearly, all this was set up to compare the spectral sequences E p,qr and F p,qr . Onthe first level we have a diagram E p,q d E −−−−−→ E p +1 ,q ∼ = ∼ =ˇ H [ p + q ] ( X p , Y p ; C ) ˇ H [ p + q +1] ( X p +1 , Y p +1 ; C ) ∼ = ∼ = (cid:81) n ∈ Z F p,q +2 n d F −−−−−→ (cid:81) n ∈ Z F p +1 ,q +2 n (2.48)Since d E ( d F ) is natural with respect to filtration-preserving morphisms of k -algebras (of presheaves over X ), these differentials must commute with the naturalisomorphisms in the diagram (2.48). This yields natural isomorphisms E p,qr ∼ = (cid:89) n ∈ Z F p,q +2 nr .2. Periodic cyclic homology of finite type algebras 27for all r ≥
1. For r = ∞ we see that there exist filtrations of finite length on HP ∗ ( A ) and ˇ H ∗ ( X ; A ), such that the associated graded objects are isomorphic.Hence HP ∗ ( A ) and ˇ H ∗ ( X ; A ), being vector spaces, are unnaturally isomorphic.Moreover they have finite dimension since every term ˇ H [ p + q ] ( X p , Y p ; C ), beingthe cohomology of the affine algebraic variety X p \ Y p , has finite dimension by [57,Theorems 4.6 and 6.1].This proves the theorem for unital finite type algebras, so let us now assumethat J is an nonunital finite type k -algebra. By stability HP ∗ ( M ( J )) ∼ = HP ∗ ( J )and the sheaves corresponding to M ( J ) and J are isomorphic, so we may assumethat J has no one-dimensional representations. Consider now the unital finite typealgebra A = k + J , with multiplication( f , b )( f , b ) = ( f f , f b + f b + b b ) (2.49)Its standard filtration is A = I st ⊃ J = I st ⊃ · · · ⊃ I stn A − ⊃ I stn A = Jac( A ) = Jac( J ) (2.50)The above considerations show that, as vector spaces, HP − m ( J ) = HP − m ( I st ) ∼ = m (cid:89) p =2 E p,m − p ∞ ∼ = m (cid:89) p =2 (cid:89) n ∈ Z F p, n + m − p ∞ ∼ = ˇ H [ m ] ( X ; I )(2.51)It only remains to see that I is isomorphic to the sheaf constructed from J , butthis is clear from looking at the stalks. (cid:50) So we managed to describe the periodic cyclic homology of a finite type k -algebra using only the following data: • the spectrum Prim( A ) with a natural topology that makes it a non-Hausdorffmanifold • the complex analytic variety X • the continuous map Θ : Prim( A ) → X For some time the author believed that this construction on page 25 couldbe extended to a cohomology theory on the category of non-Hausdorff manifolds,but now it seems to him that it only gives good results under rather restrictiveconditions. Apparently we need the following implication of (2.38) : there exists astratification of Prim( A ) such that at every level the set of non-Hausdorff points ina component is either the whole component, or a submanifold of lower dimension.8 Chapter 2. K -theory and cyclic type homology theories We would like to discuss the topological counterpart of the algebraic cyclic the-ory of Section 2.1. To prepare for this, and to fix certain notations, we startby recalling some general results for m-algebras and topological tensor products.When studying the literature, it quickly becomes clear that this topological settingis significantly more tricky than the purely algebraic setting, for several reasons.Firstly, the category of topological vector spaces is not abelian, i.e. not everyclosed subspace has a closed complement. Secondly, the tensor product of twotopological vector spaces is not unique, and the functor “ ⊗ t A ” (for some unam-biguous choice of a topological tensor product) is in general not exact. And finally,although the appropriate results are all known to experts, there does not appearto be an overview available.A topological algebra A (over C ) is an algebra with a topology such thataddition and scalar multiplication are jointly continuous, while multiplication isseparately continuous. When we talk about the spectrum of A , we usually meanthe set Prim( A ) of all closed primitive ideals of A . The closed subsets of Prim( A )are as in (2.28).A seminorm on A is a map p : A → [0 , ∞ ) with the properties • p ( λa ) = | λ | p ( a ) • p ( a + b ) ≤ p ( a ) + p ( b )for all a, b ∈ A and λ ∈ C . Moreover p is called submultiplicative if • p ( ab ) ≤ p ( a ) p ( b )We say that p (cid:48) dominates p if p (cid:48) ( a ) ≥ p ( a ) ∀ a ∈ A . If { p i } i ∈ I is a collection ofseminorms, then there is a coarsest topology on A making all the p i continuous.The sets { a ∈ A : p i ( a − b ) < /n } b ∈ A, n ∈ N , i ∈ I form a subbasis for this topology. If it agrees with the original topology, then wecall A a locally convex algebra and say that it has the topology defined by thefamily of seminorms { p i } i ∈ I . Notice that the p i may have nontrivial nullspaces N i and that A is Hausdorff if and only if ∩ i ∈ I N i = 0. Furthermore the multiplicationin A is jointly continuous if, but not only if, all the p i are submultiplicative.Two families of seminorms are equivalent if every member of either family isdominated by a finite linear combination of seminorms from the other family. Twofamilies of seminorms define the same topologies if and only if they are equivalent.A locally convex algebra is metrizable if and only if its topology can be definedby a countable family of seminorms { p i } ∞ i =1 with ∩ ∞ i =1 N i = 0. In that case a metricis given by d ( a, b ) = ∞ (cid:88) i =1 − i p i ( a − b )1 + p i ( a − b ) (2.52).3. Topological cyclic theory 29Clearly this implies a notion of completeness for such algebras, and it can begeneralized to all locally convex algebras by means of Cauchy filters on uniformspaces. For sequences this comes down to calling a sequence { a n } ∞ n =1 in A Cauchyif and only if for every i ∈ I the sequence { x n + N i } ∞ n =1 is Cauchy in the normedspace A/N i .Combining all these notions, an m-algebra is a complete Hausdorff locally con-vex algebra A whose topology can be defined by a family of submultiplicativeseminorms. We call A Fr´echet if it is metrizable on top of that. If B is a topologi-cal algebra such that GL ( B + ) is open in B + , then we call B a Q-algebra. EveryBanach algebra, but not every Fr´echet algebra, is a Q-algebra.Since m-algebras are not so well-known we state some important properties.Let A be a unital m-algebra and A × = GL ( A ) the set of invertible elements in A . Recall that the spectrum of an element a ∈ A issp( a ) = { λ ∈ C : a − λ / ∈ A × } Contrarily to the Banach algebra case, sp( a ) is in general not compact. Theorem 2.9
1. M-algebras are precisely the projective limits of Banach alge-bras.2. Inverting is a continuous map from A × to A .3. Suppose that U ⊂ C is an open neighborhood of sp ( a ) , and let C an ( U ) be thealgebra of holomorphic functions on U . There exists a unique continuousalgebra homomorphism, the holomorphic functional calculus C an ( U ) → A : f → f ( a ) such that → and id U → a .4. If Γ is a positively oriented smooth simple closed contour, around sp ( a ) andin U , then f ( a ) = 12 πi (cid:90) Γ f ( λ )( λ − a ) − dλ ∀ f ∈ C an ( U ) Proof. (cid:50)
For some typical examples, consider a C k -manifold X , with k ∈ { , , , . . . , ∞} .We shall always assume that our manifolds are σ -compact, hence in particularparacompact. Let U ⊂ R d be an open set and φ : U → X a chart. For a multi-index α with | α | = n and g ∈ C n ( U ) let ∂ α g = ∂ n g∂y α · · · ∂y α n ∈ C ( U ) (2.53)0 Chapter 2. K -theory and cyclic type homology theoriesbe the derivative of g with respect to the standard coordinates y , . . . , y d of R d .For K ⊂ U compact and n ∈ N ≤ k we define a seminorm ν n,φ,K on C k ( X ) by ν n,φ,K ( f ) = sup y ∈ K (cid:88) | α |≤ n | ∂ α ( f ◦ φ )( y ) || α | ! (2.54)Straightforward estimates show that every ν n,φ,K is submultiplicative and that C k ( X ) is complete with respect to the family of such seminorms. Moreover, be-cause X is σ -compact, we can cover it by countably many sets φ i ( K i ). { ν n,φ i ,K i : i, n ∈ N , n ≤ k } (2.55)is a countable collection of seminorms defining the topology of C k ( X ), whichtherefore is a Fr´echet algebra.Finally, if X is compact and k ∈ N then C k ( X ) is a Banach algebra. Indeed,if we cover X by finitely many sets φ i ( K i ) then (cid:107) f (cid:107) = (cid:88) i ν n,φ i ,K i ( f ) (2.56)is an appropriate norm.We now give a quick survey of topological tensor products, completely due toGrothendieck [50]. To fix the notation, we agree that by ⊗ without any sub- orsuperscript we always mean the algebraic tensor product. By default we take itover C if both factors are complex vector spaces, and over Z if there is no fieldover which both factors are vector spaces.The algebraic tensor product of two vector spaces V and W solves the universalproblem for bilinear maps. This means that every bilinear map from V × W tosome vector space Z factors as V × W −→ Z (cid:38) (cid:37) V ⊗ W resulting in a bijection between Bil( V × W, Z ) and Lin( V ⊗ W, Z ). This procedurecan be extended in several ways to the category of locally convex spaces, corre-sponding to different classes of bilinear maps and different topologies on V ⊗ W .For example we have the projective tensor product V ⊗ π W [50, SubsectionI.1.1], called so because it commutes with projective limits. It is V ⊗ W withthe topology solving the universal problem for jointly continuous bilinear maps V × W → Z . If { p i } i ∈ I and { q j } j ∈ J are defining families of seminorms for V and W , then this topology is defined by the family of seminorms γ ij ( x ) = inf (cid:40) n (cid:88) k =1 p i ( v k ) q j ( w k ) : x = n (cid:88) k =1 v k ⊗ w k (cid:41) i ∈ I, j ∈ J (2.57)The completion V (cid:98) ⊗ W of V ⊗ W for the associated uniform structure is called thecompleted projective tensor product..3. Topological cyclic theory 31Similarly the inductive tensor product V ⊗ i W [50, Subsection I.3.1] solves theuniversal problem for separately continuous bilinear maps, and it commutes withinductive limits. The topology of V ⊗ i W is finer than that of V ⊗ π W , and theassociated completion is denoted by V ⊗ W . Typically, for a C k -manifold X and aBanach space V we have C k ( X ) ⊗ V ∼ = C k ( X ; V ) (2.58)There exists also more subtle structures on V ⊗ W , such as the injective tensorproduct V ⊗ (cid:15) W [50, p. I.89], which in a certain sense has the weakest reasonabletopology.If V satisfies V ⊗ (cid:15) Z = V ⊗ π Z for every Z then it is called nuclear, and ifboth V and W are nuclear, then so are V ⊗ π W and V (cid:98) ⊗ W . On the other hand,if V and W are both Fr´echet spaces, then V ⊗ i W = V ⊗ π W [50, p. I.74] and itscompletion V ⊗ W = V (cid:98) ⊗ W is again a Fr´echet space [50, Th´eor`eme II.2.2.9].Consequently the tensor powers of a nuclear Fr´echet space can be defined un-ambiguously. For example if X and Y are smooth manifolds, then C ∞ ( X ) and C ∞ ( Y ) are nuclear Fr´echet spaces and C ∞ ( X ) (cid:98) ⊗ C ∞ ( Y ) ∼ = C ∞ ( X × Y ) (2.59)Now that we have come this far, it is logical to spend a few words on topologicaltensor products over rings. So let A be an m-algebra, V a right A -module and W a left A -module. We assume that V and W are complete Hausdorff locally convexspaces and that the module operations are jointly continuous. Then the completedprojective tensor product V (cid:98) ⊗ A W is the completion of V ⊗ A W for the topologysolving the universal problem for jointly continuous A -bilinear maps from V × W to some A -module Z . Just as over C , this topology is defined by the family ofseminorms (2.57).Let us return to homology of algebras. In any category of locally convex alge-bras with a topological tensor product ⊗ t we can form the bicomplex CC per ( A, ⊗ t )with spaces CC perp,q ( A, ⊗ t ) = A ⊗ t q +1 The maps from (2.3) and (2.4) are continuous because they use only the algebraoperations of A . This, and the subcomplexes CC ( A, ⊗ t ) and CC { } ( A, ⊗ t ), lead tofunctors HH n ( A, ⊗ t ) , HC n ( A, ⊗ t ) and HP n ( A, ⊗ t ). They are related by Connes’periodicity exact sequence, but to get more nice features it is imperative that weuse only completed tensor products and place ourselves in one of the followingcategories: • CLA : complete Hausdorff locally convex algebras • MA : m-algebras • FA : Fr´echet algebras • BA : Banach algebras2 Chapter 2. K -theory and cyclic type homology theoriesAlthough the objects of none these categories form a set, we allow ourselves touse ∈ to indicate with what kind of algebra we are dealing.By Theorem 2.9 the completed projective tensor product of two m-algebrasis again an m-algebra, so we use (cid:98) ⊗ as our default and ⊗ as a reserve. Just asin Section 2.1 we are going to study the functorial properties of the resultinghomology theories. Let A, A m ∈ CLA , m ∈ N .Notice that the topological cyclic bicomplexes under consideration contain thealgebraic cyclic bicomplexes. This yields natural transformations from the alge-braic cyclic theories to their topological counterparts. Therefore any homomor-phism from a complex algebra B to A induces maps on homology groups like HH n ( B ) → HH n ( A, (cid:98) ⊗ )These maps are compatible with all the properties below.1. Additivity. HH n (cid:32) ∞ (cid:77) m =1 A m , ⊗ (cid:33) ∼ = ∞ (cid:77) m =1 HH n ( A m , ⊗ ) HH n (cid:32) ∞ (cid:89) m =1 A m , (cid:98) ⊗ (cid:33) ∼ = ∞ (cid:89) m =1 HH n ( A m , (cid:98) ⊗ )and similarly for HC n . The corresponding isomorphisms for ⊗ and (cid:81) hold if A m ∈ FA ∀ m and the isomorphisms for (cid:98) ⊗ and (cid:76) are valid if A m ∈ BA ∀ m .For HP n we can only be sure about the case with (cid:81) and (cid:98) ⊗ .The proof of all these statements can be reduced to that of the algebraiccase, by using [50, Propositions I.1.3.6 and I.3.1.14].2. Stability. HH n ( M m ( A ) , (cid:98) ⊗ ) ∼ = HH n ( A, (cid:98) ⊗ )and similarly with HC n , HP n and ⊗ .This follows from the algebraic case, since all topological tensor products of A with a finite dimensional vector space (such as M m ( C )) are the same, andessentially equal to the algebraic tensor product.It is not known to the author whether HH n and HC n are Morita-invariantin a more general sense, but for HP n we will soon return to this point.3. Continuity. Here great concessions to the algebraic case must be made. As-sume that all the A m are nuclear Fr´echet algebras and that A = lim m →∞ A m is a strict inductive limit. (Strict means that all the maps A m → A m +1 areinjective and have closed range.) In this setting Brodzki and Plymen showed[16, Theorem 2] that HH n ( A, ⊗ ) ∼ = lim m →∞ HH n ( A m , ⊗ ) HC n ( A, ⊗ ) ∼ = lim m →∞ HC n ( A m , ⊗ ).3. Topological cyclic theory 33To make HP n continuous we need even more conditions. For example if ∃ N ∈ N such that HH n ( A m , ⊗ ) = 0 ∀ n > N, ∀ m , then by [16, Theorem 3] HP n ( A, ⊗ ) ∼ = lim m →∞ HP n ( A m , ⊗ )The author knows of no continuity results for (cid:98) ⊗ , which is not surprising, con-sidering the bad compatibility of projective tensor products with inductivelimits.Excision is also pretty subtle for topological algebras. Let A be one of thefour categories from page 31. Extending Wodzicki’s terminology, we call A ∈ A strongly H-unital if, for every V ∈ A , the homology of the differential gradedcomplex ( A b ⊗ n (cid:98) ⊗ V , b (cid:48) (cid:98) ⊗ id V ) is 0.It follows from [66, Section 1] that every Banach algebra with a left or rightbounded approximate identity (e.g. a C ∗ -algebra) is strongly H-unital, and in [15,Section 3] it is claimed that this also holds in FA .Recall that an extension of topological vector spaces 0 → Y → Z → W → Y (or more precisely, its image) has a closed complement in Z . Furthermore we callan extension 0 → A → B → C → A topologically pure if, for every V ∈ A ,0 → A (cid:98) ⊗ V → B (cid:98) ⊗ V → C (cid:98) ⊗ V → FA :1. admissible extensions2. extensions (2.60) such that A has a bounded left or right approximate iden-tity3. extensions of nuclear Fr´echet algebrasWith this terminology, the following is proved in [15, Theorems 2 and 4] : Theorem 2.10
Let → A → B → C be a topologically pure extension of Fr´echetalgebras, with A strongly H-unital. Then there are long exact sequences → HH n ( A, (cid:98) ⊗ ) → HH n ( B, (cid:98) ⊗ ) → HH n ( C, (cid:98) ⊗ ) → HH n − ( A, (cid:98) ⊗ ) →→ HC n ( A, (cid:98) ⊗ ) → HC n ( B, (cid:98) ⊗ ) → HC n ( C, (cid:98) ⊗ ) → HC n − ( A, (cid:98) ⊗ ) →→ HP n ( A, (cid:98) ⊗ ) → HP n ( B, (cid:98) ⊗ ) → HP n ( C, (cid:98) ⊗ ) → HP n − ( A, (cid:98) ⊗ ) → With the help of Theorem 2.9, all these results on excision (except 3.) can beextended to the category of m-algebras.4 Chapter 2. K -theory and cyclic type homology theoriesActually HP ∗ has much more features than those listed above. Let f, g : A → B be morphisms in CLA . We say that they are homotopic if there exists a morphism φ : A → C ([0 , , B ) such that f = ev ◦ φ and g = ev ◦ φ . They are calleddiffeotopic if there exists a morphism φ : A → C ∞ ([0 , (cid:98) ⊗ B ∼ = C ∞ ([0 , B )with these properties. Theorem 2.11
In the category MA the functor HP ∗ ( · , (cid:98) ⊗ ) has the followingproperties:1. If f, g are diffeotopic, then HP ∗ ( f ) = HP ∗ ( g ) .2. Let E and F be linear subspaces of an m-algebra A , and let A ( EF ) , respec-tively A ( F E ) , be the subalgebra generated by all the products ef , respectively f e , with e ∈ E , f ∈ F . Then HP ∗ ( A ( EF )) ∼ = HP ∗ ( A ( F E )) .3. Every admissible extension (2.60) gives rise to an exact hexagon HP ( A, (cid:98) ⊗ ) → HP ( B, (cid:98) ⊗ ) → HP ( C, (cid:98) ⊗ ) ↑ ↓ HP ( C, (cid:98) ⊗ ) ← HP ( B, (cid:98) ⊗ ) ← HP ( A, (cid:98) ⊗ ) Proof. (cid:50)
A clear omission at this point is a K¨unneth theorem for topological periodiccyclic homology. It certainly exists, but the author does not know in what gen-erality. Fortunately, for all the algebras that we use there is an ad hoc argumentavailable to prove the K¨unneth isomorphism.What happens to differential forms in the presence of a topology? If A is acommutative unital m-algebra, then the definition of Ω ( A ) must be modified toretain completeness. So, identifying the K¨ahler differential a db with the elemen-tary tensor a ⊗ b , we define Ω ( A, (cid:98) ⊗ ) to be the quotient of A (cid:98) ⊗ A by the closed A -submodule generated by the relations (2.16). Furthermore let V n be closed sub-space of (cid:0) Ω ( A, (cid:98) ⊗ ) (cid:1) b ⊗ A n generated by all the n -forms ω ∧ · · · ∧ ω n for which thereexist i (cid:54) = j with ω i = ω j . ThenΩ n ( A, (cid:98) ⊗ ) = (cid:94) nA Ω ( A, (cid:98) ⊗ ) := (cid:0) Ω ( A, (cid:98) ⊗ ) (cid:1) b ⊗ A n /V n (2.62)Thus, finally, we have the topological De Rham homology H DRn ( A, (cid:98) ⊗ ) = H n (cid:0) Ω ∗ ( A, (cid:98) ⊗ ) , d (cid:1) (2.63)Let us consider the topological counterpart of a smooth algebra. It is not exactlyclear what that should be, but obviously it should be related to algebras of smooth.4. Topological K -theory and the Chern character 35functions. Nuclearity is also an advantage. So let C ∞ ( X ) be the (nuclear Fr´echet)algebra of infinitely often differentiable complex valued functions on a smooth realmanifold X . It is well known that in this case we have natural isomorphismsΩ n ( C ∞ ( X ) , (cid:98) ⊗ ) ∼ = Ω n ( X ) (2.64) H DRn ( C ∞ ( X ) , (cid:98) ⊗ ) ∼ = H nDR ( X ) (2.65)These hold both with real and with complex coefficients, but we are mostly in-terested in the latter. Furthermore the maps (2.20) and the Hochschild-Kostant-Rosenberg theorem can be extended to this topological situation [32, 126, 134], sothere are natural isomorphisms HH n ( C ∞ ( X ) , (cid:98) ⊗ ) ∼ = Ω n ( X ) (2.66) HC n ( C ∞ ( X ) , (cid:98) ⊗ ) ∼ = Ω n ( X ) /d Ω n − ( X ) ⊕ H n − DR ( X ) ⊕ H n − DR ( X ) ⊕ · · · (2.67) HP n ( C ∞ ( X ) , (cid:98) ⊗ ) ∼ = (cid:89) m ∈ Z H n +2 mDR ( X ) (2.68)We conclude the section with a warning. An algebra may be “too big” for cyclictheory to work properly. In fact the results are pretty noninformative for mostBanach algebras. Let A be an amenable Banach algebra [66], for example C ( Y )with Y a compact Hausdorff space or L ( G ) with G a locally compact amenablegroup. Then we have HH n ( A, (cid:98) ⊗ ) = (cid:26) A/ [ A, A ] if n = 00 if n > A, A ] is the range of the commutator map A (cid:98) ⊗ A → A . Thus ∀ n ≥ HC n ( A, (cid:98) ⊗ ) = HP n ( A, (cid:98) ⊗ ) = A/ [ A, A ] HC n +1 ( A, (cid:98) ⊗ ) = HP n +1 ( A, (cid:98) ⊗ ) = 0 (2.70) K -theory and the Chern charac-ter Topological K -theory is at the very heart of noncommutative geometry. For a com-pact topological space it is defined roughly speaking as the Grothendieck group ofequivalence classes of vector bundles over X . By the Gelfand-Na˘ımark and Serre-Swan theorems it can be transferred to (commutative) C ∗ -algebras, and there itbecomes the Grothendieck group of isomorphism classes of finitely generated pro-jective modules. This in turn can be extended to Banach algebras, and on thatcategory K ∗ ( A ) is something like the Grothendieck group of homotopy classes ofidempotents or invertibles in the stabilization of A .6 Chapter 2. K -theory and cyclic type homology theoriesIn the present section we study the K -functor on even larger categories oftopological algebras. We collect some important theorems, focussing especiallyon those results that are fit to compare K -theory with topological periodic cyclichomology.Of course there also exists a purely algebraic K -theory, which is a naturalcompanion of the algebraic cyclic theory of Section 2.1. However, since thesealgebraic K -groups are notoriously difficult to compute, and since they containmore number-theoretic than geometric information, we will not study them here.The most general construction of a topological K -functor is due to Cuntz [33],and it realizes K ∗ as the covariant half of a bivariant functor on the category ofm-algebras. As Cuntz’s construction is rather complicated, we will not elaborateon it. Instead we recall the definition of Phillips [102], which works for Fr´echetalgebras and is similar to that for Banach algebras.Let K be the nuclear Fr´echet algebra of infinite matrices with rapidly decreasingcoefficients. It is also referred to as the algebra of smooth compact operators,because it is a holomorphically closed dense *-subalgebra of the usual algebra ofcompact operators, and it is isomorphic, as a nuclear Fr´echet space, to the algebraof smooth functions C ∞ ( T ) on the two-dimensional torus.For any Fr´echet algebra A , let ( K (cid:98) ⊗ A ) + be the unitization of K (cid:98) ⊗ A , and considerthe Fr´echet algebra M (cid:0) ( K (cid:98) ⊗ A ) + (cid:1) . Define ¯ P ( A ) to be the set of all idempotents e in this algebra satisfying e − (cid:18) (cid:19) ∈ M ( K (cid:98) ⊗ A )Similarly ¯ U ( A ) is the set of all invertible elements u ∈ M (cid:0) ( K (cid:98) ⊗ A ) + (cid:1) for which u − (cid:18) (cid:19) ∈ M ( K (cid:98) ⊗ A )Following [102, Definition 3.2] we put K ( A ) = π (cid:0) ¯ P ( A ) (cid:1) (2.71) K ( A ) = π (cid:0) ¯ U ( A ) (cid:1) (2.72)With the multiplication defined by the direct sum of matrices, these turn out tobe abelian groups with unit elements (cid:20)(cid:18) (cid:19)(cid:21) and (cid:20)(cid:18) (cid:19)(cid:21) Later we shall want to pick “nice” representants of K -theory classes, so now wetry to discover how much is possible in this respect. Let A be unital, e ∈ M n ( A )idempotent and u ∈ GL n ( A ). Pick a rank one projector p ∈ K and an isomorphism M n ( K ) → K and extend it to λ n : M n ( K (cid:98) ⊗ A ) ∼ −−→ K (cid:98) ⊗ A .4. Topological K -theory and the Chern character 37Now consider the elements (cid:18) λ n ( pep ) (cid:19) ∈ ¯ P ( A ) and (cid:18) λ n (1 − p + pup ) (cid:19) ∈ ¯ U ( A ) (2.73)The resulting classes in K ∗ ( A ) do not depend on p and λ n , and are simply denoted[ e ] and [ u ]. The natural inclusion u → u ⊕ GL n ( A ) in GL n +1 ( A ) enables usto construct the inductive limit grouplim n →∞ π (cid:0) GL n ( A ) (cid:1) Similarly, the inclusion e → e ⊕ M n ( A ) in M n +1 ( A ) leads to the inductivelimit space K +0 ( A ) := lim n →∞ π (cid:0) Idem M n ( A ) (cid:1) Actually this is an abelian semigroup with unit element 0. By [102, Lemma 7.4] itis naturally isomorphic to the monoid of equivalence classes of finitely generatedprojective A -modules. In this notation [102, Theorem 7.7] becomes Theorem 2.12
Let A be a unital Fr´echet Q-algebra. The assignments e → [ e ] and u → [ u ] extend to natural isomorphisms G (cid:0) K +0 ( A ) (cid:1) ∼ −−→ K ( A )lim n →∞ π (cid:0) GL n ( A ) (cid:1) ∼ −−→ K ( A ) where the G stands for Grothendieck group. In particular K ( A ) has a natural ordering, for which K +0 ( A ) is precisely thesemigroup of positive elements. These construction are especially important inconnection with density theorem for K -theory [12, Th´eor`eme A.2.1] : Theorem 2.13
Let A and B be Fr´echet Q-algebras, and φ : A → B a morphismwith dense range. Suppose that a ∈ A + is invertible whenever φ + ( a ) ∈ B + isinvertible. Then for any n ∈ N the induced maps Idem M n ( A + ) → Idem M n ( B + ) GL n ( A + ) → GL n ( B + ) are homotopy equivalences, and K ∗ ( φ ) : K ∗ ( A ) → K ∗ ( B ) is an isomorphism. The conditions are typically satisfied if B is a unital Banach algebra, A is adense unital subalgebra which is Fr´echet in its own finer topology, and A ∩ B × = A × .If we are working in m*-algebras then everywhere in the above discussion wemay replace invertibles by unitaries, and idempotents by projections. This is aconsequence of the following elementary result.8 Chapter 2. K -theory and cyclic type homology theories Lemma 2.14
Let A be a unital m*-algebra such that sp ( z ∗ z ) ⊂ R + ∀ z ∈ A . Theset of unitaries in A is a deformation retract of the set of invertibles in A . Likewise,the set of projections in A is a deformation retract of the set of idempotents in A .Proof. Using Theorem 2.9.3 write | z | = ( z ∗ z ) / . Then z | z | − is unitary for every z ∈ A × and [0 , × A × → A × : ( t, z ) → z | z | − t is the desired deformation retraction. Similarly, there is a natural path from anidempotent to its associated Kaplansky projector, see e.g. [10, Proposition 4.6.2]. (cid:50) Quite often it is possible to find a bound on the size n of matrices that we needto construct all K -classes. To measure this we recall the notion of topologicalstable rank. Given a unital topological algebra A define Lg n ( A ) := { ( a , . . . , a n ) ∈ A n : Aa + · · · + Aa n = A } (2.74) tsr ( A ) := inf { n : Lg n ( A ) is dense in A n } (2.75)Rieffel [106, 107] showed that this is useful for K -theory of C ∗ -algebras. The mostgeneral result in this direction is [107, Theorem 2.10] : Theorem 2.15
Let A be a unital C ∗ -algebra. For any n ≥ tsr ( A ) we have π (cid:0) GL n ( A ) (cid:1) ∼ = K ( A )To bound the topological stable rank of algebras that are not too far from com-mutative we use the following tools, cf. [106, Propostion 1.7] and [100, Theorem2.4] : Proposition 2.16
Let X be a compact Hausdorff space and dim X its coveringdimension. Also let A ⊂ B be an inclusion of unital C ∗ -algebras, such that B is aleft A -module of rank n . Then tsr ( C ( X )) = 1 + (cid:98) dim X/ (cid:99) tsr ( B ) ≤ n tsr ( A )Together with Theorems 2.12 - 2.15 this will allow us to realize the K -groupof certain C ∗ -algebras entirely by invertible matrices, of a certain bounded size,with coefficients in a dense subalgebra.Now we return to the study of the more abstract features of the K -functor.1. Additivity. For any m-algebras A m ( m ∈ N ) K n (cid:32) ∞ (cid:89) m =1 A m (cid:33) ∼ = ∞ (cid:89) m =1 K n ( A m ).4. Topological K -theory and the Chern character 392. Stability. K n ( K (cid:98) ⊗ A ) ∼ = K n ( M m ( A )) ∼ = K n ( A )3. Continuity. If A m ( m ∈ N ) are Banach algebras and A = lim m →∞ A m istheir Banach inductive limit, then K n ( A ) ∼ = lim m →∞ K n ( A m )4. Excision. Let 0 → A → B → C → K ( A ) → K ( B ) → K ( C ) ↑ ↓ K ( C ) ← K ( B ) ← K ( A )5. Diffeotopy invariance. Let f, g : A → B be diffeotopic morphisms of m-algebras, or homotopic morphisms of Fr´echet algebras. Then K ∗ ( f ) = K ∗ ( g )In this list 3 is classical, but the author does not know of any extension toFr´echet algebras. Proofs of 1, 2, 4 and 5 can be found in [33] and [102].Obviously we will compare the features of K ∗ with those of HP ∗ given inSection 2.3. Since topological K -theory is built with the completed projectivetensor product, it only makes sense to compare it with the cyclic theory withthe same topological tensor product. Hence, from now on HP ∗ ( A ) will mean HP ∗ ( A, (cid:98) ⊗ ) for any m-algebra A , unless explicitly specified otherwise.First we deduce from the excision and diffeotopy properties that K ∗ and HP ∗ react in the same way on suspending an algebra. This is essentially a manifestationof Bott periodicity. By definition the smooth suspension of A is S ∞ ( A ) = { f ∈ C ∞ ( S ; A ) : f (1) = 0 } Lemma 2.17
For i = 0 , there are natural isomorphisms K − i ( A ) ∼ −−→ K i ( S ∞ ( A )) K ( A ) ⊕ K ( A ) ∼ −−→ K i ( C ∞ ( S ; A )) The same holds for periodic cyclic homology.Proof.
Let (not an algebra! but easy to repair) C ∞ ( A ) := { f ∈ C ∞ ([0 , A ) : f (0) = 0 , f ( n ) (0) = f ( n ) (1) ∀ n > } be the smooth cone of A . Consider the admissible extension0 → S ∞ ( A ) → C ∞ ( A ) → A → K -theory and cyclic type homology theorieswhere the third map is evaluation at 0 and the second arrow comes from composinga function with the surjection[0 , → S : x → e πix The boundary maps from the exact hexagon associated with (2.76) are the desiredmaps K − i ( A ) → K i ( S ∞ ( A )). To prove that these are isomorphisms we will showthat C ∞ ( A ) is diffeotopy equivalent to the algebra 0. Let r : [0 , → [0 ,
1] be abijective diffeomorphism with the properties • r (cid:48) ( x ) > ∀ x ∈ (0 , • r ( n ) (0) = r ( n ) (1) = 0 ∀ n > C ∞ ( A ) → C ∞ ([0 , , { } ; A ) : f → fC ∞ ([0 , , { } ; A ) → C ∞ ( A ) : f → f ◦ r Since r is diffeotopic to id [0 , , both compositions of these algebra homomorphismsare diffeotopic to the respective identity homomorphisms. Hence we get naturalisomorphisms K ∗ ( C ∞ ( A )) ∼ −−→ K ∗ ( C ∞ ([0 , , { } ; A ))However, C ∞ ([0 , , { } ; A ) is diffeotopy equivalent to 0 by means of the homo-morphisms φ t : C ∞ ([0 , , { } ; A ) → C ∞ ([0 , , { } ; A ) φ t ( f )( s ) = f ( ts )and therefore its K -theory vanishes.Similarly there is an admissible extension0 → S ∞ ( A ) → C ∞ ( S ; A ) → A → a ∈ A to the element ˜ a ∈ C ∞ ( S ; A ) with˜ a ( t ) = a ∀ t ∈ S . Applying K i we get a split exact sequence of abelian groups0 → K i ( S ∞ ( A )) → K i ( C ∞ ( S ; A )) → K i ( A ) → K − i ( A ) ⊕ K i ( A ) ∼ −−→ K i ( S ∞ ( A )) ⊕ K i ( A ) ∼ −−→ K i ( C ∞ ( S ; A ))The same proof applies with periodic cyclic homology. (cid:50) Continuing our comparison, we see from Theorem 2.11.2 that HP ∗ is also K -stable. Namely, we may take for E all the matrices whose only nonzero entries are.4. Topological K -theory and the Chern character 41in the first column, and for F all the matrices which have only zeros outside thefirst row. The isomorphism HP ∗ ( A ) ∼ −−→ HP ∗ ( K (cid:98) ⊗ A ) (2.78)is induced by the algebra morphism a → pap , where p ∈ K is an arbitrary rankone projector. Its inverse HP ∗ ( K (cid:98) ⊗ A ) ∼ −−→ HP ∗ ( A ) (2.79)is a little more tricky, since it is not given by an algebra morphism, but a morphismof bicomplexes, the so-called generalized trace map. This is the linear map tr : (cid:0) K (cid:98) ⊗ A (cid:1) b ⊗ n → A b ⊗ n (2.80)defined on elementary tensors by tr ( k a ⊗ · · · ⊗ k n a n ) = tr ( k · · · k n ) a ⊗ · · · ⊗ a n (2.81)Note that this works equally well if K is replaced by a finite dimensional matrixalgebra M m ( C ), cf. [81, Section 1.2].So now we know that HP ∗ is halfexact, diffeotopy invariant and K -stable. Onthe other hand, Cuntz’ kk [33, Section 6] is the universal halfexact, diffeotopyinvariant, K -stable bivariant functor from m-algebras to abelian groups. Thisimplies the existence of a unique natural transformation of functors ch : K ∗ → HP ∗ (2.82)respecting these features. It is called the Chern character, because it is a far-reaching generalization of the classical Chern character Ch : K ∗ ( X ) → ˇ H ∗ ( X ; Q ) (2.83)that assigns to a complex vector bundle over a paracompact Hausdorff space X aclass in the even ˇCech cohomology. Indeed, we can get (2.83) for smooth manifoldsby applying (2.82) to C ∞ ( X ) and using the isomorphism (2.68).The Chern character is compatible with the countable additivity of K ∗ and HP ∗ , and also with excision, as was shown by Nistor [95, Theorem 1.6] : Theorem 2.18
Let → A → B → C → be an extension of m-algebras. Thevarious Chern characters make a commutative diagram K ( A ) → K ( B ) → K ( C ) → K ( A ) → K ( B ) → K ( C ) ↓ ↓ ↓ ↓ ↓ ↓ HP ( A ) → HP ( B ) → HP ( C ) → HP ( A ) → HP ( B ) → HP ( C ) Moreover, if the extension is admissible and η : K ( C ) → K ( A ) and ∂ : HP ( C ) → HP ( A ) denote the connecting maps, then ch ◦ η = 2 πi ∂ ◦ ch . K -theory and cyclic type homology theoriesExplicit formulas for the Chern character, from Phillips’ picture to the cyclicbicomplex, were first given by Karoubi [69, Chapitre II]. In the setting of (2.73)we may replace A by M n ( A ) to achieve that e, u ∈ A . Define c m +1 ( e ) = ( − m (2 m )! m ! ( e, . . . , e ) ∈ A ⊗ m +1 c m ( e ) = ( − m − (2 m )!2( m !) ( e, . . . , e ) ∈ A ⊗ m c m ( u ) = ( m − u − , u, . . . , u − , u ) ∈ A ⊗ m c m +1 ( u ) = m ! (2 , u − , u, . . . , u − , u ) ∈ A ⊗ m +1 (2.84)and place c n ( e ) in CC per − n,n − ( A, (cid:98) ⊗ ) ∼ = A b ⊗ n and c n ( u ) in CC per − n,n − ( A, (cid:98) ⊗ ) ∼ = A b ⊗ n .By Lemma 2.1.6, Theorem 8.3.4 and Proposition 8.4.9 of [81] we have ch [ e ] = [( c n ( e )) ∞ n =1 ] ∈ HP ( A ) (2.85) ch [ u ] = [( c n ( u )) ∞ n =1 ] ∈ HP ( A ) (2.86)With the density theorem and the homotopy invariance of K -theory we cancompute it for many Fr´echet algebras, in particular commutative ones. The max-imal ideal space of a commutative m-algebra A is defined like in algebraic geome-try : it is the collection Max( A ) of all closed maximal ideals of A , endowed withthe coarsest topology that makes all elements of A into continuous functions onMax( A ). This is called the Gelfand topology, and we denote it by T G .Contrarily to the C ∗ -algebra case, there may be several commutative Fr´echetalgebras with the same maximal ideal space. The spectrum of a commutativeFr´echet algebra is Hausdorff, σ -compact and paracompact, but it need not belocally compact. Therefore we also consider the compactly generated topology T c on Max( A ). This means that we call U ⊂ Max( A ) open in T c if and only if U ∩ C is open in C , for any compact C with the relative topology from (Max( A ) , T G ). If A + is the unitization of A , then Max( A + ) = Max( A ) ∪ { A } and we put C A := { f ∈ C (Max( A + ) , T c ) : f ( A ) = 0 } (2.87) Theorem 2.19
For any commutative Fr´echet algebra A there are natural isomor-phisms K ∗ ( A ) ∼ = K ∗ ( C A ) ∼ = K ∗ (cid:0) Max( A + ) , { A } (cid:1) If Max ( A ) is locally compact then K ∗ ( A ) ⊗ Q ∼ = ˇ H ∗ (Max( A + ) , { A } ) ⊗ Q Proof.
The isomorphisms with integral coefficients are due to Phillips [102, The-orem 7.15]. Here K ∗ means representable K -theory of topological spaces, in thesense of Karoubi [68]. He represents K n ( X ) := [ X, F U n ] (2.88)as the set of homotopy classes of continuous maps from a paracompact Hausdorffspace X to some classifying space F U n . This agrees with the usual definition if.4. Topological K -theory and the Chern character 43 X is compact, but in general it yields a generalized cohomology theory withoutcompact supports. It has been known since the beginning of topological K -theorythat (2.83) gives an isomorphism Ch ⊗ id Q : K ∗ ( X ) ⊗ Q ∼ −−→ ˇ H ∗ ( X ) ⊗ Q (2.89)if X is a finite CW-complex [2, Section 2.4]. With spectral sequences, as in [114],one can extend this to all compact Hausdorff spaces, since these are homotopyequivalent to CW-complexes. (cid:50) This theorem can be considered as the counterpart in topological K -theory ofTheorem 2.6. If we apply it to a smooth manifold we get K ∗ ( C ∞ ( X )) ⊗ C ∼ = ˇ H ∗ ( X ; C ) (2.90)Since ˇCech cohomology agrees with De Rham cohomology (both with complexcoefficients) on the category of smooth manifolds, we deduce from (2.68), (2.90)and the naturality of the Chern character that ch ⊗ id C : K ∗ ( C ∞ ( X )) ⊗ C ∼ −−→ HP ∗ ( C ∞ ( X )) (2.91)is an isomorphism. If we think a little more about this, it becomes clear that suchan isomorphism should hold for many more algebras, even noncommutative ones.To make this precise, we introduce yet another category of topological algebras,denoted CIA . It is a full subcategory of the category of m-algebras MA , and itsobjects are those A ∈ MA for which the Chern character induces an isomorphism ch ⊗ id C : K ∗ ( A ) ⊗ C ∼ −−→ HP ∗ ( A ) (2.92) Proposition 2.20
The category
CIA is closed under the following operations:1. countable direct products2. tensoring with M m ( C ) or K
3. diffeotopy equivalences4. admissible extensions, quotients and idealsProof. K ∗ and HP ∗ on pages 32and 39. As concerns 4, by Theorem 2.18 we can apply Lemma 2.3 to the functors K ∗ ( · ) ⊗ C and HP ∗ ( · , (cid:98) ⊗ ) on the category MA with admissible morphisms. Thefactor 2 πi in Theorem 2.18 is inessential. (cid:50) In view of these similarities, it is logical to try to extend the material fromSection 2.2 to topological K -theory. However, this is somewhat problematic, as4 Chapter 2. K -theory and cyclic type homology theoriesgeneral compact Hausdorff spaces are much less easy to handle then algebraicvarieties. In the next section we will avoid these difficulties by considering onlysmooth manifolds with a finite group action. Right now we will prove a coarseanalogue of Theorem 2.7, which applies to semisimple algebras which ”live” onfinite simplicial complexes. We formulate it in terms of C ∗ -algebras, but withTheorem 2.13 it can easily be generalized to certain Fr´echet algebras. Proposition 2.21
Let Σ be a finite simplicial complex and φ : A → B a homo-morphism of C ∗ -algebras. Suppose that • there are unital homomorphisms from C (Σ) to the centers of the multiplieralgebras M ( A ) and M ( B ) • for every simplex σ of Σ there are finite dimensional C ∗ -algebras A σ and B σ such that AC ( σ, δσ ) ∼ = C ( σ, δσ ; A σ ) and BC ( σ, δσ ) ∼ = C ( σ, δσ ; B σ ) • φ is C (Σ) -linear • for every x σ ∈ σ \ δσ the localization φ ( x σ ) : A σ → B σ induces an isomor-phism on K -theoryThen K ∗ ( φ ) : K ∗ ( A ) ∼ −−→ K ∗ ( B ) is an isomorphism.Proof. Let Σ n be the n -skeleton of Σ and consider the ideals C (Σ) = I ⊃ I ⊃ · · · ⊃ I n ⊃ · · · I n = C (Σ , Σ n ) (2.93)They give rise to ideals A n = AI n and B n = BI n . Because Σ is finite all theseideals are 0 for large n . We can identify A n − /A n ∼ = AC (Σ n , Σ n − ) ∼ = (cid:77) σ ∈ Σ : dim σ = n AC ( σ, δσ ) := (cid:77) σ ∈ Σ : dim σ = n C ( σ, δσ ; A σ )and similarly for B . Because φ is C (Σ)-linear, it induces homomorphisms φ ( σ ) : C ( σ, δσ ; A σ ) → C ( σ, δσ ; B σ )By Lemma 2.3 and the additivity of K -theory it suffices to show that every φ ( σ )induces an isomorphism on K -theory. Let x σ be any interior point of σ . Because σ \ δσ is contractible, φ σ is homotopic to id C ( σ,δσ ) ⊗ φ ( x σ ). By assumption thelatter map induces an isomorphism on K -theory. With the homotopy invarianceof K -theory it follows that K ∗ ( φ ( σ )) is an isomorphism. (cid:50) Note that this proof applies equally well to the functor K ∗ ( · ) ⊗ Z Q ..5. Equivariant cohomology and algebras of invariants 45 This section is all about algebras that carry an action of a finite group, and theirsubalgebras of invariant elements. To place things in a classical context we firstrecall some beautiful theorems on equivariant topological K -theory and cyclictheory for crossed product algebras.All the results after that are due to the author, but some of them alreadyappeared in [123]. We broaden our view, to algebras of the form A G = C ∞ ( X ; M N ( C )) G with X a smooth manifold. We relate K -theory of such algebras to a G -equivariantcohomology theory due to Bredon [14], which can also be described as the ˇCechcohomology of a certain sheaf over the orbifold X/G . This depends on the exis-tence of a G -equivariant triangulation of X . Using the same Mayer-Vietoris typeof arguments we also prove that the Chern character for A G becomes an isomor-phism after tensoring with C . Finally, if X happens to be a complex affine variety,then we show that the ”polynomial” subalgebra A Galg has the same periodic cyclichomology as A G .Let G be a topological group acting continuously on a topological space X .Then G also acts continuously on the closed subspace (cid:101) X := { ( g, x ) ∈ G × X : gx = x } (2.94)of G × X by g ( g (cid:48) , x ) = ( gg (cid:48) g − , gx ) (2.95)and (cid:101) X/G is called the extended quotient of X by G . In the literature one oftenencounters the notation ˆ X for (cid:101) X , but we avoid this because it might be confusedwith the spectrum of a topological group.Let (cid:104) G (cid:105) be the set of conjugacy classes in G , and denote the class containing g by (cid:104) g (cid:105) . We have a decomposition (cid:101) X/G ∼ = (cid:71) (cid:104) g (cid:105)∈(cid:104) G (cid:105) ( g, X g /Z G ( g )) ∼ = (cid:71) (cid:104) g (cid:105)∈(cid:104) G (cid:105) X g /Z G ( g ) (2.96)where Z G ( g ) is the centralizer of g in G and X g = { x ∈ X : gx = x } (2.97)Notice that the components of this partition are always closed, and they are openif G is finite.A G -vector bundle over X is a vector bundle p : V → X together with anaction of G on V , such that ∀ v ∈ V, x ∈ X, g ∈ G K -theory and cyclic type homology theories • p ( gv ) = gp ( v ) • g : p − ( x ) → p − ( gx ) is linearIf X and G are both compact Hausdorff, then it makes sense to consider theGrothendieck group of equivalence classes of complex G -vector bundles. Thisgroup was first studied by Atiyah [1], and it is denoted by K G ( X ). By the samesuspension procedure as in the nonequivariant case this leads to a sequence of func-tors K nG , together called equivariant K -theory. This is a equivariant cohomologytheory which shares most of the properties of ordinary topological K -theory.For any g ∈ G , the restriction of a complex G -bundle p : V → X to X g isa vector bundle on which g acts linearly in every fiber. So we can decompose itcanonically into to its g -eigenspaces : V (cid:12)(cid:12) X g = (cid:77) i V i := (cid:77) i (cid:8) v ∈ p − ( X g ) : gv = λ i v (cid:9) (2.98)By the continuity of the action and the compactness of X g there are only finitelymany λ i ∈ C for which V i is nonzero. From this decompostion we cook a canonicalmap ρ g : K ∗ G ( X ) → K ∗ ( X g ) ⊗ C (2.99)sending [ V ] to (cid:80) i λ i [ V i ]. All these ρ g ’s together combine to a map that classifies G -bundles over X in terms of ordinary vector bundles over the extended quotient (cid:101) X/G . Indeed, for finite G the identification K ∗ ( (cid:101) X ) ∼ = (cid:77) (cid:104) g (cid:105)∈(cid:104) G (cid:105) K ∗ ( X g ) (2.100)gives a map ρ := (cid:88) g ∈ G ρ g : K ∗ G ( X ) → K ∗ ( (cid:101) X ) ⊗ C (2.101)It is easy to see that the image of ρ is contained in the subspace of G -invariants,so if we compose it with the Chern character for (cid:101) X we land in ˇ H ∗ ( (cid:101) X ; C ) G , whichby [51, Corollaire 5.2.3] is naturally isomorphic to ˇ H ∗ ( (cid:101) X/G ; C ). By the way, thiscomposition Ch G := ( Ch ⊗ id C ) ◦ ρ (2.102)is called the equivariant Chern character. The punchline is of course [4, Theorem1.19] : Theorem 2.22
For any finite group G acting on a compact Hausdorff space X there are natural isomorphisms ρ ⊗ id C : K ∗ G ( X ) ⊗ C ∼ −−→ (cid:16) K ∗ ( (cid:101) X ) ⊗ C (cid:17) G Ch G ⊗ id C : K ∗ G ( X ) ⊗ C ∼ −−→ (cid:16) ˇ H ∗ ( (cid:101) X ; C ) (cid:17) G ∼ = ˇ H ∗ ( (cid:101) X/G ; C ).5. Equivariant cohomology and algebras of invariants 47We switch back to a more algebraic point of view. Suppose that the compactgroup G acts by *-automorphisms on a C ∗ -algebra A . The above leads us toconsider finitely generated projective A -modules M with a G -action satisfying g ( am ) = ( ga )( gm ). The Grothendieck group of equivalence classes of such modulesis the equivariant K -theory K G ( A ). This gives rise to sequence of functors K Gn which by the equivariant Serre-Swan theorem [101, Theorem 2.3.1] are related tothe above homonymous functors as K G ∗ ( C ( X )) ∼ = K ∗ G ( X ) (2.103)We already managed to describe G -bundles in terms of ordinary vector bundlesover a related space, and the same is possible for C ∗ -algebras. Namely, Julg [67]showed that there is a natural identification K G ∗ ( A ) ∼ = K ∗ ( A (cid:111) G ) (2.104)Combining (2.103) and (2.104) with Theorem 2.22 we see that K ∗ ( C ( X ) (cid:111) G ) ⊗ C ∼ = ˇ H ∗ ( (cid:101) X ; C ) G ∼ = ˇ H ∗ ( (cid:101) X/G ; C ) (2.105)This is important for our purposes, since it shows that the homology of a crossedproduct algebra can be described in geometric terms. This can even be refined incyclic theory. Let X be either a nonsingular complex affine variety or a smoothreal manifold, not necessarily compact, and A the algebra of either regular orsmooth functions on X . Suppose that the action of G preserves this structure, sothat the partially invariant subspaces X g are of the same type as X . Brylinski[21, 22] proved that HH n ( A (cid:111) G ) ∼ = (cid:77) (cid:104) g (cid:105)∈(cid:104) G (cid:105) Ω n ( X g ) Z G ( g ) ∼ = Ω n (cid:0) (cid:101) X (cid:1) G (2.106) HP n ( A (cid:111) G ) ∼ = (cid:77) (cid:104) g (cid:105)∈(cid:104) G (cid:105) (cid:89) m ∈ Z H n +2 mDR ( X g ) Z G ( g ) ∼ = (cid:89) m ∈ Z H n +2 mDR (cid:0) (cid:101) X (cid:1) G (2.107) HC n ( A (cid:111) G ) ∼ = (cid:77) (cid:104) g (cid:105)∈(cid:104) G (cid:105) (cid:16) Ω n ( X g ) /d Ω n − ( X g ) ⊕ H n − DR ( X g ) ⊕ H n − DR ( X g ) ⊕ · · · (cid:17) Z G ( g ) (2.108) ∼ = (cid:16) Ω n (cid:0) (cid:101) X (cid:1) /d Ω n − (cid:0) (cid:101) X (cid:1) ⊕ H n − DR (cid:0) (cid:101) X (cid:1) ⊕ H n − DR (cid:0) (cid:101) X (cid:1) ⊕ · · · (cid:17) G Moreover Nistor [96, Theorem 2.11] constructed an explicit map HH n ( A (cid:111) G ) → (cid:77) (cid:104) g (cid:105)∈(cid:104) G (cid:105) Ω n ( X g ) (2.109)(not the naive restriction!) and showed that it induces these isomorphisms.When we combine (2.105),(2.107) and Theorem 2.13 with the naturality of theChern character, we arrive at8 Chapter 2. K -theory and cyclic type homology theories Theorem 2.23
Suppose that a finite group G acts by diffeomorphisms on a com-pact smooth manifold X . Then the Chern character gives an isomorphism ch ⊗ id C : K ∗ ( C ∞ ( X ) (cid:111) G ) ⊗ C ∼ −−→ HP ∗ ( C ∞ ( X ) (cid:111) G )Later we will see that the compactness assumption in this theorem is notnecessary, so we drop it now, at least for the rest of this section.Interestingly, the crossed product C ∞ ( X ) (cid:111) G can also be realized as an algebraof invariants : C ∞ ( X ) (cid:111) G ∼ = C ∞ (cid:0) X ; End( C [ G ]) (cid:1) G (2.110)where the group algebra C [ G ] carries the right regular representation of G , seeLemma A.3. We propose to study such algebras also with other G -representationsinstead of C [ G ]. Then C [ G ] is universal in the sense that it contains every irre-ducible G -representation. At the other extreme we have the trivial representation,which leads us to the Fr´echet algebra C ∞ ( X/G ) := C ∞ ( X ) G (2.111)of smooth functions on the orbifold X/G , cf. [110]. Wassermann [134, SectionIV] showed that the periodic cyclic homology of this algebra equals, as one wouldexpect, the ˇCech cohomology of
X/G : HP ∗ (cid:0) C ∞ ( X ) G (cid:1) ∼ = H ∗ DR ( X ) G ∼ = ˇ H ∗ ( X/G ; C ) (2.112)But he also noticed that Hochschild homology does not behave so well in this case,as HH ∗ (cid:0) C ∞ ( X ) G (cid:1) and Ω ∗ ( X ) G are not isomorphic in general.Let Z ⊂ Y be arbitrary subsets of R n , and V a Fr´echet space. To includemanifolds with boundary in our studies we adhere to the following conventions : C ∞ ( Y ) := (cid:8) f (cid:12)(cid:12) Y : f ∈ C ∞ ( U ) for some open U with Y ⊂ U ⊂ R n (cid:9) C ∞ ( Y, Z ) := (cid:8) f ∈ C ∞ ( Y ) : f (cid:12)(cid:12) Z = 0 (cid:9) C ∞ ( Y, Z ; V ) := C ∞ ( Y, Z ) (cid:98) ⊗ V (2.113)Unfortunately this slightly ambiguous for orbifolds embedded in R n . For exampleif Y = R / {± } , identified as a topological space with [0 , ∞ ), then C ∞ (cid:0) [0 , ∞ ) (cid:1) (cid:41) C ∞ ( R ) {± } since the right hand side contains only functions whose odd derivatives vanishat 0. However, the difference is not too big, since both algebras are diffeotopyequivalent to C ⊕ C ∞ (cid:0) R , ( −∞ , (cid:1) via f → f ◦ φ , where φ ∈ C ∞ (cid:0) R , ( −∞ , (cid:1) isan automorphism of [0 , ∞ ) which is diffeotopic to the identity of [0 , ∞ ). In suchsituations we shall usually give priority to the orbifold structure and use (2.111)as a definition, at least locally..5. Equivariant cohomology and algebras of invariants 49Now let X be a smooth manifold with boundary, still σ -compact, and considerthe Fr´echet algebra A := C ∞ ( X ; M N ( C )) ∼ = M N ( C ∞ ( X )) (2.114)We assume we have elements u g ∈ A × and diffeomorphisms α g of X such that ga ( x ) = u g ( x ) a ( α − g x ) u − g ( x ) (2.115)defines an action of G on A . Although this implies that g → α g is a grouphomomorphism, g → u g need not be one. The algebra A G = C ∞ ( X ; M N ( C )) G (2.116)will be our Fr´echet version of a finite type algebra. Clearly A G is finitely generatedas a module over C ∞ ( X ) G . It follows from a classical theorem of Newman thatthe set of points of X whose G -stabilizer equals ker α is open and everywheredense in X , see [42, Theorem 1]. Hence, if ker α = { e } then Z ( A G ) = C ∞ ( X ) G .To compute its K -theory we will use an “ equivariant cohomology theory witha local coefficient system”, as defined by Bredon [14]. This theory can be combinedwith the ideas of Segal [115] and S(cid:32)lomi´nska [120] to describe K ∗ (cid:0) A G (cid:1) in sheaf-cohomological terms.First we recall some of Bredon’s constructions, referring to [14] for more preciseinformation. Let Σ be a countable, locally finite and finite dimensional G -CWcomplex. Assume that all cells are oriented and that the action of G preservesthese orientations.We define a category K whose objects are the finite subcomplexes of Σ. Themorphisms from K to K (cid:48) are the maps K → K (cid:48) : x → gx for g ∈ G such that gK ⊂ K (cid:48) . Now a local coefficient system L on Σ is a covariant functor from K to the category of abelian groups, and the group C q (Σ; L ) of q -cochains is theset of all functions f on the q -cells of Σ with the property that f ( τ ) ∈ L ( τ ) ∀ τ .Furthermore we define a coboundary map ∂ : C q (Σ; L ) → C q +1 (Σ; L ) by( ∂f )( σ ) = (cid:88) τ [ τ : σ ] L ( τ → σ ) f ( τ ) (2.117)where the sum runs over all q -cells τ and the incidence number [ τ : σ ] is the degreeof the attaching map from ∂σ (the boundary of σ in the standard topologicalsense) to τ /∂τ . The group G acts naturally on this complex by cochain maps, so,for any K ⊂ Σ , (cid:0) C ∗ ( K ; L ) G , ∂ (cid:1) is a differential complex and we can define theequivariant cohomology of K with coefficients in L as H qG ( K ; L ) := H q (cid:0) C ∗ ( K ; L ) G , ∂ (cid:1) (2.118)More generally for K (cid:48) ⊂ K, C ∗ ( K, K (cid:48) ; L ) is the kernel of the restriction map C ∗ ( K ; L ) → C ∗ ( K (cid:48) ; L ) and H qG ( K, K (cid:48) ; L ) = H q (cid:0) C ∗ ( K, K (cid:48) ; L ) G , ∂ (cid:1) (2.119)0 Chapter 2. K -theory and cyclic type homology theoriesBy construction there exists a local coefficient system L G (more or less consisting ofthe G -invariant elements of L ) on the CW-complex Σ /G such that the differentialcomplexes ( C ∗ ( K, K (cid:48) ; L ) G , ∂ ) and ( C ∗ ( K/G, K (cid:48) /G ; L G ) , ∂ ) are isomorphic. Noticethat L G defines a sheaf over Σ /G (with the cells as cover), so that H qG ( K, K (cid:48) ; L ) ∼ = ˇ H q (cid:0) K/G, K (cid:48) /G ; L G (cid:1) (2.120)Let Σ p be the p -skeleton of Σ. We capture all the above things in a spectralsequence ( E p,qr ) r ≥ , degenerating already for r ≥
2, as follows : E p,q = H p + qG (Σ p , Σ p − ; L ) = (cid:26) C p (Σ; L ) G if q = 00 if q > E p,q = (cid:26) H pG (Σ; L ) if q = 00 if q > d E is the composition E p,q → H p + qG (Σ p ; L ) → E p +1 ,q (2.123)of the maps induced by the inclusion (Σ p , ∅ ) → (Σ p , Σ p − ) and the coboundary ∂ .Now let B G be an algebra like (2.116), but without the differentiable structure.We will see later that this algebra has the same K -theory as A G , for a suitabletriangulation of X . So we put B = C (Σ; M N ( C )) = M N ( C (Σ)) (2.124)and we assume that we have u g ∈ B × such that gb ( x ) = u g ( x ) b ( g − x ) u − g ( x ) (2.125)defines an action of G on B . To associate a local coefficient system L u to thisalgebra we first assume that K is connected. In that case we let G K := { g ∈ G : gx = x ∀ x ∈ K } (2.126)be the isotropy group of K and we define L u ( K ) to be the free abelian group onthe (equivalence classes of) irreducible projective G K -representations contained in( π x , C N ), where π x ( g ) = u g ( x ) for g ∈ G K , x ∈ K . By the continuity of the u g weget the same group for any x ∈ K . If K is not connected, then we let { K i } i be itsconnected components, and we define L u ( K ) = (cid:89) i L u ( K i ) (2.127)Suppose that gK ⊂ K (cid:48) and that ρ is a projective G K -representation. Then wedefine a projective G K (cid:48) -representation by L u ( g : K → K (cid:48) ) ρ ( g (cid:48) ) = ρ ( g − g (cid:48) g ) g (cid:48) ∈ G K (cid:48) (2.128).5. Equivariant cohomology and algebras of invariants 51If h ∈ G gives the same map from K to K (cid:48) as g then h − g ∈ G K and L u ( h : K → K (cid:48) ) ρ ( g (cid:48) ) = ρ ( h − g (cid:48) h ) = ρ ( h − g ) ρ ( g − g (cid:48) g ) ρ ( g − h ) (2.129)so L u ( h : K → K (cid:48) ) ρ is isomorphic to L u ( g : K → K (cid:48) ) ρ as a projective representa-tion. This makes L u into a functor.Suppose for example that u g ( x ) = 1 ∀ x ∈ Σ , g ∈ G . Then L u and L Gu are theconstant sheaves Z over Σ and Σ /G respectively, and H ∗ G (Σ; L u ) ∼ = ˇ H ∗ (Σ /G ; Z ) (2.130)is the ordinary cellular cohomology of Σ /G . Moreover B G ∼ = C (Σ /G ; M N ( C )), so K ∗ (cid:0) B G (cid:1) ∼ = K ∗ (Σ /G ), which is isomorphic to ˇ H ∗ (Σ /G ; Z ) modulo torsion.Of most interest is the case where B G ∼ = C (Σ) (cid:111) G is the crossed product, asin Lemma A.3. Then we compare L Gu to the direct image of the constant sheaf Z on (cid:101) Σ, under the canonical map p : (cid:101) Σ /G → Σ /G . Although they may not alwaysbe isomorphic, their ˇCech complexes are identical, for any cover that refines thecell structure. Since p is finite to one we can deduce, using even more spectralsequences [47, Chapitre 5], that H ∗ G (Σ; L u ⊗ Q ) ∼ = ˇ H ∗ (cid:0)(cid:101) Σ /G ; Q (cid:1) (2.131)and by Theorem 2.23 this is the same as K ∗ (cid:0) B G (cid:1) ⊗ Q if Σ is compact, i.e. if it isa finite CW-complex.It turns out that this close relation between K ∗ (cid:0) B G (cid:1) and the ˇCech cohomology H ∗ (cid:0) Σ /G ; L Gu (cid:1) is valid in general. Consider the following analogue of (2.33) K ∗ (cid:0) B G (cid:1) = K ∗ (cid:0) B G (cid:1) ⊃ K ∗ (cid:0) B G (cid:1) ⊃ · · · ⊃ K dim Σ ∗ (cid:0) B G (cid:1) ⊃ K ∗ (cid:0) B G (cid:1) = 0 K p ∗ (cid:0) B G (cid:1) := ker (cid:0) K ∗ (cid:0) C (Σ; M N ( C )) G (cid:1) → K ∗ (cid:0) C (Σ p − ; M N ( C )) G (cid:1)(cid:1) (2.132) Theorem 2.24
The graded group associated with the filtration (2.132) is isomor-phic to ˇ H ∗ (cid:0) Σ /G ; L Gu (cid:1) . In particular there is an (unnatural) isomorphism K ∗ (cid:0) B G (cid:1) ⊗ Q ∼ = ˇ H ∗ (cid:0) Σ /G ; L Gu ⊗ Q (cid:1) (2.133) and K ∗ (cid:0) B G (cid:1) ∼ = ˇ H ∗ (cid:0) Σ /G ; L Gu (cid:1) if one of both sides is torsion free.Proof. Using [26, Section XV.7] we construct a spectral sequence ( F p,qr ) q ∈ Z / Z r ≥ with the following terms: F p,q = K p + q (cid:0) C (Σ p / Σ p − ; M N ( C )) G (cid:1) F p,q = H p (Σ /G ; K qu ) F p,q ∞ = K pp + q (cid:0) B G (cid:1) /K p +1 p + q (cid:0) B G (cid:1) (2.134)2 Chapter 2. K -theory and cyclic type homology theorieswhere K qu ( σ ) = K ∗ (cid:0) C ( Gσ ; M N ( C )) G (cid:1) . Now replace L in (2.121) by L u and sumover all q . If we compare the result with F p = F , ⊕ F , we see that E p ∼ = F p .So we get a diagram like (2.48) : F p,q d F −−−−−→ F p +1 ,q ∼ = ∼ = (cid:81) n ∈ Z E p,q +2 n d E −−−−−→ (cid:81) n ∈ Z E p +1 ,q +2 n (2.135)The differential d F for F is induced from the construction of a mapping cone ofa Puppe sequence in the category of C ∗ -algebras. This is the noncommutativecounterpart of the construction of the differential in cellular homology, so by natu-rality d F corresponds to d E under the above isomorphism. Therefore the spectralsequences E pr and F pr are isomorphic, and in particular F r degenerates for r ≥ K ∗ (cid:0) B G (cid:1) or ˇ H ∗ (cid:0) Σ /G ; L Gu (cid:1) is torsion free, then every term E p,q ∞ ∼ = F p,q ∞ must be torsion free. Hence in this case both K ∗ (cid:0) B G (cid:1) and ˇ H ∗ (cid:0) Σ /G ; L Gu (cid:1) are freeabelian groups, of the same rank. (cid:50) The main use of this theorem is really to compute K ∗ (cid:0) B G (cid:1) , for now the ex-tensive machinery of ˇCech cohomology becomes available.The requirements (2.126) - (2.128) allow us to construct the sheaves L u and L Gu without reference to the cellular structure of Σ. If we do this for the algebra A G of(2.116) then the stalk of L u over x ∈ X is the free abelian group on the (equivalenceclasses of) irreducible projective G x -representations contained in ( π x , C N ) and the G -action on L u is determined by (2.128). A section s is continuous at x if thereexists a neighborhood U of x such that ∀ y ∈ U : • G y ⊂ G x • s ( x ) = s ( y ) as virtual projective G y -representationsThis L u is a generalization of the sheaf constructed in [4, § L Gu of G -invariant continuous sections descends to a sheaf on X/G .To relate this sheaf to the K -theory and periodic cyclic homology of A G weneed two preparatory results. The first is a weak version of Theorem 2.19, whichhowever does include HP ∗ . Lemma 2.25
Let U ⊂ R n be an open bounded star-shaped set. The Fr´echetalgebra C ∞ ( R n , R n \ U ) belongs to CIA and K ∗ ( C ∞ ( R n , R n \ U )) ∼ = ˇ H ∗ ( R n , R n \ U ; Z ) ∼ = Z is concentrated in degree n . .5. Equivariant cohomology and algebras of invariants 53 Proof.
Clearly we may assume that 0 is the center of U . Let P be the point of the n -sphere corresponding to infinity under the stereographic projection S n → R n .By assumption C ∞ ( R n , R n \ U ) ∼ = C ∞ ( S n , Y ) for some closed neighborhood Y of P , and we will show that the latter algebra is diffeotopy equivalent to C ∞ ( S n , P ).Let ( r t ) t ∈ [0 , be a diffeotopy of smooth maps S n → S n such that • ∀ t : r t ( P ) = P and r t ( Y ) ⊂ Y • a neighborhood of − P is fixed pointwise by all r t • r = id S n and r ( Y ) = P To construct such maps, we can require that r t stabilizes every geodesic from − P to P and declare that furthermore r t ( Q ) depends only on t and on the distancefrom Q to P . Then we only have to pick a suitable smooth function of t and thisdistance. Given this, consider the Fr´echet algebra homomorphisms φ : C ∞ ( S n , P ) → C ∞ ( S n , Y ) φ ( f ) = f ◦ r i : C ∞ ( S n , Y ) → C ∞ ( S n , P ) i ( f ) = f (2.136)By construction φ ◦ i and i ◦ φ are diffeotopic to the respective identity mapson C ∞ ( S n , Y ) and C ∞ ( S n , P ), so these algebras are indeed diffeotopy equivalent.Thus we reduced our task to calculating the K -groups and periodic cyclic homologyof C ∞ ( S n , P ). Fortunately there is an obvious split extension0 → C ∞ ( S n , P ) → C ∞ ( S n ) → C → CIA . It is well known that K ∗ ( C ∞ ( S n )) ∼ = K ∗ ( S n ) ∼ = ˇ H ∗ ( S n ; Z ) ∼ = Z (2.138)with one copy of Z in degree 0 and the other in degree n . Since K ∗ ( C ) = K ( C ) ∼ = Z the lemma follows from the excision property of the K -functor. (cid:50) Next we prove an equivariant version of the Poincar´e lemma.
Lemma 2.26
Let
X, A, G and A G be as in (2.116) , and suppose that X is G -equivariantly contractible to a point x ∈ X . Then A G is diffeotopy equivalent toits fiber End G (cid:0) C N (cid:1) over x . In particular K ∗ (cid:0) A G (cid:1) = K (cid:0) A G (cid:1) is a free abeliangroup of finite rank, and A G ∈ CIA .Proof. Our main task is to adjust the u g suitably. Since X is contractible wecan find for every g ∈ G a function f g ∈ C ∞ ( X ) such that f − Ng = det( u g ). The G -action does not change when we replace u g with f g u g , so we may assume that4 Chapter 2. K -theory and cyclic type homology theoriesdet( u g ) ≡ , ∀ g ∈ G . The premise that (2.115) is a group action guarantees thatthere is a smooth function λ : G × G × X → C such that u g ( x ) u h ( α − g x ) = λ ( g, h, x ) u gh ( x ) (2.139)Taking determinants we see that in fact λ ( g, h, x ) N ≡
1, so λ does not dependon x ∈ X . All the elements of α ( G ) fix x , so the fiber V = C N over that pointcarries a projective G -representation π . Thus we are in a position to apply Schur’stheorem [113], which says that there exists a finite central extension G ∗ of G suchthat π lifts to a representation of G ∗ . This lift only involves scalar multiples ofthe u g ( x ), so it immediately extends to X . Then (2.139) becomes the cocyclerelation u gh ( x ) = u g ( x ) u h ( α − g x ) (2.140)Notice that still A G ∗ = A G , so without loss of generality we can replace G by G ∗ .Now we want to make the u g ( x ) independent of x ∈ X . Wassermann [133]indicated how this can be done in the continuous case, and his argument caneasily be adapted to our smooth setting. The crucial observation, first made byRosenberg [109], is that A G can be rewritten as the image of an idempotent in alarger algebra. This idempotent can then be deformed to one independent of x .Indeed, let A (cid:111) α G be the crossed product of A and G with respect to theaction α of G on X , and ( r t ) t ∈ [0 , a smooth G -equivariant contraction from X to x . (For smooth manifolds the existence of a continuous contraction implies theexistence of a smooth one.) Define p t ( x ) := | G | − (cid:88) g ∈ G u g ( r t x ) g (2.141)Then p t ∈ A (cid:111) α G is an idempotent by (2.140), and by Lemma A.2 φ : A G → p ( A (cid:111) α G ) p φ ( σ ) = p σp (2.142)is an isomorphism of Fr´echet algebras. Clearly the idempotents p t are all homo-topic, so they are conjugate in the completion C ( X ; M N ( C )) (cid:111) α G of A (cid:111) α G , whichis a Banach algebra if X is compact. Moreover the standard argument for this, asfor example in [10, Proposition 4.3.2], shows that p and p are conjugate by anelement of A (cid:111) α G . Alternatively we can use the stronger result that homotopicidempotents in Fr´echet algebras are conjugate, but this statement is vastly moredifficult to prove than its Banach algebra version, cf. [102, Lemmas 1.12 and 1.15].In any case, we have A G ∼ = p ( A (cid:111) α G ) p ∼ = p ( A (cid:111) α G ) p ∼ = C ∞ ( X ; End C ( V )) G (2.143)To this last algebra we can apply the obvious diffeotopy σ → σ ◦ r t . This showsthat A G is diffeotopy equivalent to its fiber End G ( V ) over x , and the remainingstatements on K ∗ (cid:0) A G (cid:1) and HP ∗ (cid:0) A G (cid:1) follow from the semisimplicity of the finite.5. Equivariant cohomology and algebras of invariants 55dimensional algebra End G ( V ) . (cid:50) Now we can prove the main result of this section, which extends Lemma 2.26to general X . Theorem 2.27
Let
X, A, G and A G = C ∞ ( X ; M N ( C )) G be as in (2.116) , and let L Gu be the sheaf over X/G constructed on page 52. Then there exists a filtrationon K ∗ (cid:0) A G (cid:1) whose associated graded group is isomorphic to ˇ H ∗ (cid:0) X/G ; L Gu (cid:1) , andthe Chern character induces an isomorphism K ∗ (cid:0) A G (cid:1) ⊗ C ∼ −−→ HP ∗ (cid:0) A G (cid:1) Moreover K ∗ (cid:0) A G (cid:1) is a finitely generated abelian group whenever X is compact.Proof. All our arguments will depend on the existence of a specific cover of X . Toconstruct it we use a theorem of Illman [62], which states that X has a smoothequivariant triangulation. In slightly more down-to-earth language this means(among others) that there exists a countable, locally finite simplicial complex Σ ina finite dimensional orthogonal representation space V of G , and a G -equivarianthomeomorphism ψ : Σ → X . Moreover ψ is smooth as a map from a subset of V to X , and its restriction to any simplex σ of Σ is an embedding. In particular Σis a G -CW complex, so the assertion on the ˇCech cohomology of L Gu follows fromTheorems 2.13 and 2.24.For a simplex σ we put U (cid:48) σ := { v ∈ Σ : d ( v, σ ) ≤ r σ } (2.144)where d is the Euclidean distance in V . We require that the radius r σ dependsonly on the G -orbit of σ and that r τ > r σ > τ is a face of σ . The orthogonalityof the action of G on V guarantees that gU (cid:48) σ = U (cid:48) gσ and U (cid:48) σ ∩ U (cid:48) τ ⊂ U (cid:48) σ ∩ τ if we take our radii small enough. Let D (cid:48) σ be the union, over all faces τ of σ , ofthe U (cid:48) τ , and G σ the stabilizer of σ in G . From the above we deduce that U (cid:48) σ \ D (cid:48) σ is G σ -equivariantly retractible to σ \ D (cid:48) σ .Now we abbreviate U σ := ψ ( U (cid:48) σ ) and D σ := ψ ( D (cid:48) σ ), so that { U σ : σ simplex ofΣ } is a closed G -equivariant cover of X . Let X m be the union of all those U σ forwhich m + dim σ ≤ dim X . It is a closed subvariety (with boundary and corners)of X and it is stable under the action of G . Define the following G -stable idealsof A : I m := { a ∈ A : a (cid:12)(cid:12) X m = 0 } (2.145)By [128, Th´eor`eme IX.4.3]0 → I m → A = C ∞ ( X ; M N ( C )) → C ∞ ( X m ; M N ( C )) → K -theory and cyclic type homology theoriesis an admissible extension of Fr´echet algebras. Using the finiteness of G we seethat I Gm is an admissible ideal in I Gm +1 and that I Gm +1 /I Gm ∼ = ( I m +1 /I m ) G ∼ = C ∞ ( X m , X m +1 ; M N ( C )) G (2.147)In order to apply Lemma 2.3 to the sequence0 = I G ⊂ I G ⊂ · · · ⊂ I G dim X ⊂ I G X = A G (2.148)we only have to show that the algebras in (2.147) are in the category CIA . In fact,since U σ \ D σ ∩ U τ \ D τ = ∅ if dim σ = dim τ and σ (cid:54) = τ , we have an isomorphism I m +1 /I m ∼ = (cid:89) m +dim σ =dim X C ∞ ( U σ , D σ ; M N ( C )) (2.149)Now G permutes the simplices in this product, so I Gm +1 /I Gm ∼ = (cid:89) σ ∈ L m C ∞ ( U σ , D σ ; M N ( C )) G σ (2.150)where L m is a set of representatives of the simplices of dimension dim X − m modulo the action of G . Invoking the additivity of K ∗ and HP ∗ we reduce ourtask to verifying that every factor of (2.150) belongs to CIA .If m = dim X then D σ is empty and we see from Lemma 2.26 that C ∞ ( U σ ; M N ( C )) G σ has the required property.For smaller m there also exists (for every σ ) a G σ -equivariant contraction of U σ to a point x σ ∈ ψ ( σ ). Thus we can follow the proof of Lemma 2.26 up to equation(2.143), where we find that the factor of (2.150) corresponding to σ is isomorphic to C ∞ ( U σ , D σ ; End C ( V σ )) G σ . Here ( π σ , V σ ) denotes the projective G σ -representationover the point x σ . Using the G σ -equivariant retraction of U σ \ D σ to ψ ( σ \ D (cid:48) σ ) wesee that this algebra is diffeotopy equivalent to C ∞ ( σ, σ ∩ D (cid:48) σ ) ⊗ End G σ ( V σ ). Theright hand side of this tensor product has finite dimension and is semisimple, soby the stability of CIA it presents no problems. Seen from its barycenter σ \ D (cid:48) σ isstar-shaped, hence by Lemma 2.25 the left hand side is also in the category CIA .We conclude that all the algebras in (2.147) and (2.150) are indeed objects of
CIA , so Lemma 2.3 can be applied to (2.148) to prove that A G ∈ CIA .Note that the simplicial complex Σ has only finitely many vertices if X is com-pact, so then all the above direct products are in fact finite and K ∗ (cid:0) A G (cid:1) is finitelygenerated. (cid:50) It is clear from the proofs of Lemma 2.26 and Theorem 2.27 that many similarFr´echet algebras are also in
CIA . For example if Y is a closed submanifold of X then the algebra B = { f ∈ C ∞ ( X ; M ( C )) : f ( y ) diagonal ∀ y ∈ Y } (2.151)is in CIA , as can be seen from the admissible extension0 → C ∞ ( X, Y ; M ( C )) → B → C ∞ ( Y ) → C ∞ ( Y ), with Y an orbifold. Although it is not unlikely that these are all in thecategory CIA , it seems that a substantial generalization of Lemma 2.26 is neededto show this.Since the periodic cyclic homology of A G is finite dimensional and can becomputed in terms of ˇCech cohomology, it is not surprising that the K¨unnethformula is an isomorphism for such algebras Corollary 2.28
Let ( X, A, G, u ) and ( X (cid:48) , A (cid:48) , G (cid:48) , u (cid:48) ) both be sets of data like weused in (2.116) . There is a natural isomorphism of graded vector spaces HP ∗ (cid:0) A G (cid:1) ⊗ HP ∗ (cid:0) A (cid:48) G (cid:48) (cid:1) ∼ −−→ HP ∗ (cid:0) A G (cid:98) ⊗ A (cid:48) G (cid:48) (cid:1) Proof.
By Theorem 2.27 we have HP ∗ (cid:0) A G (cid:1) ∼ = ˇ H ∗ (cid:0) X/G ; L Gu ⊗ Z C (cid:1) HP ∗ (cid:0) A (cid:48) G (cid:48) (cid:1) ∼ = ˇ H ∗ (cid:0) X (cid:48) /G (cid:48) ; L G (cid:48) u (cid:48) ⊗ Z C (cid:1) HP ∗ (cid:0) A G (cid:98) ⊗ A (cid:48) G (cid:48) (cid:1) ∼ = ˇ H ∗ (cid:16) ( X × X (cid:48) ) / ( G × G (cid:48) ); ( L Gu ⊗ Z L G (cid:48) u (cid:48) ) G × G (cid:48) ⊗ Z C (cid:17) ∼ = ˇ H ∗ (cid:16) X/G × X (cid:48) /G (cid:48) ; ( L Gu ⊗ Z C ) ⊗ C ( L G (cid:48) u (cid:48) ⊗ Z C ) (cid:17) (2.153)According to [47, § C ∗ (cid:0) X/G ; L Gu ⊗ Z C (cid:1) ⊗ C C ∗ (cid:0) X (cid:48) /G (cid:48) ; L G (cid:48) u (cid:48) ⊗ Z C (cid:1) −→ ˇ C ∗ (cid:16) X/G × X (cid:48) /G (cid:48) ; ( L Gu ⊗ Z C ) ⊗ C ( L G (cid:48) u (cid:48) ⊗ Z C ) (cid:17) Because all three cohomology groups are finite dimensional vector spaces the ab-stract K¨unneth theorem [26, Theorem VI.3.1] tells us that HP ∗ (cid:0) A G (cid:1) ⊗ HP ∗ (cid:0) A (cid:48) G (cid:48) (cid:1) ∼ = HP ∗ (cid:0) A G (cid:98) ⊗ A (cid:48) G (cid:48) (cid:1) The construction of the map Θ in [44, p. 211] is also possible in the topologicalsetting, and yields a natural mapΘ : HP ∗ (cid:0) A G (cid:1) ⊗ HP ∗ (cid:0) A (cid:48) G (cid:48) (cid:1) → HP ∗ (cid:0) A G (cid:98) ⊗ A (cid:48) G (cid:48) (cid:1) Although the isomorphisms (2.153) are not natural, they come from certain fil-trations of the underlying topological spaces, which is enough to ensure that Θ isalso an isomorphism. (cid:50)
Notice also the similarity between L Gu and the sheaf A contructed on page25. Suppose that our manifold X has the underlying structure of a complexnonsingular affine variety X alg , that the α g are automorphisms of X alg and thatthe u g are invertibles in A alg := O ( X alg ) ⊗ M N ( C ) = M N ( O ( X alg )) (2.154)8 Chapter 2. K -theory and cyclic type homology theoriesThen ga ( x ) = u g ( x ) a ( α − g x ) u − g ( x ) (2.155)defines an action of G on A alg , and A Galg has the same irreducible representationsas A G . Applying the recipe on page 25 to the finite type algebra A Galg , we seethat the bases of the stalks L Gu ( Gx ) and A ( Gx ) can be identified, and that therequirements for continuity of sections on page 25 reduce to those on page 52.Therefore A = L Gu ⊗ C (2.156)This insight, together with Theorems 2.8 and 2.27, was the inspiration for anexplicit comparison theorem between algebraic and topological periodic cyclic ho-mology, analogous to (2.26). Theorem 2.29
The inclusion A Galg → A G induces an isomorphism of finite di-mensional vector spaces HP ∗ (cid:0) A Galg (cid:1) ∼ −−→ HP ∗ (cid:0) A G (cid:1) Proof.
After noticing that by Theorem 2.8 the left hand side has finite dimen-sion, we introduce some notations. Let Y be any complex algebraic variety, Z asubvariety and V a complex vector space. Like for smooth functions we write O ( Y, Z ) = { f ∈ O ( Y ) : f (cid:12)(cid:12) Z = 0 } (2.157) O ( Y, Z ; V ) = O ( Y, Z ) ⊗ V (2.158)Start with the finite collection L of all irreducible components of the X galg , as g runs over G . Extend this to a collection { V j } j of subvarieties of X alg by includingall irreducible components of the intersection of any subset of L . Notice thatdim (cid:0) X g ∩ X h (cid:1) < max (cid:8) dim X g , dim X h (cid:9) (2.159)if α g (cid:54) = α h . Define G -stable subvarieties X p = (cid:91) j :dim V j ≤ p V j (2.160)and construct the ideals I p := { a ∈ A Galg : a ( X p ) = 0 } ∼ = O ( X, X p ; M N ( C )) G J p := { a ∈ A G : a ( X p ) = 0 } ∼ = C ∞ ( X, X p ; M N ( C )) G (2.161)From (2.146) and (2.147) we see that all the ideals J p are admissible in A G , andby Theorem 2.27 they are in CIA . By Lemma 2.3 it suffices to show that for every p the inclusion O ( X p , X p − ; M N ( C )) G ∼ = I p − /I p → C ∞ ( X p , X p − ; M N ( C )) G ∼ = J p − /J p (2.162).5. Equivariant cohomology and algebras of invariants 59induces an isomorphism on periodic cyclic homology. These algebras have thesame primitive ideal spectrum, namely Y p \ Z p , where Y p = Prim (cid:0) A G /J p (cid:1) = Prim (cid:16) A Galg /I p (cid:17) Z p = Prim (cid:0) A G /J p − (cid:1) = Prim (cid:16) A Galg /I p − (cid:17) (2.163)Because all the representations ( π x , C N ) are completely reducible A Galg = I ⊃ I ⊃ · · · ⊃ I N = 0 (2.164)is an abelian filtration of A Galg , in the sense of [77, Definition 3], and Z (cid:0) A Galg /I p (cid:1) ∩ I p − /I p = Z ( I p − /I p ) ∼ = O ( Y p , Z p ) (2.165)This gives alternative descriptions Y p = Max (cid:0) Z (cid:0) A Galg /I p (cid:1)(cid:1) (2.166) Z p = { I ∈ Y p : Z ( I p − /I p ) ⊂ I } (2.167)and the proof of [77, Theorem 10] shows that there are natural isomorphisms HP ∗ ( I p − /I p ) ∼ = HP ∗ ( Z ( I p − /I p )) ∼ = ˇ H ∗ ( Y p , Z p ; C ) (2.168)To get something similar on the topological side we turn to K -theory, knowingalready that J p − /J p ∈ CIA . Moreover this algebra is dense in C ( X p , X p − ; C ) G ,so from theorem 2.13 we get HP ∗ ( J p − /J p ) ∼ = K ∗ (cid:0) C ( X p , X p − ; C ) G (cid:1) ⊗ C (2.169)Since u g ∈ A alg the type of ( π x , C N ) as a projective G x -representation cannotchange along the (connected or irreducible) components of X G x ∩ X p \ X p − . Itfollows from this, (2.159) and Theorem 2.24 that the inclusions C ∞ ( Y p , Z p ) → C ( Y p , Z p ) ∼ = Z (cid:0) C ( X p , X p − ; C ) G (cid:1) → C ( X p , X p − ; C ) G (2.170)induce isomorphisms on K -theory with rational coefficients. Moreover K ∗ ( C ∞ ( Y p , Z p )) ⊗ C ∼ = HP ∗ ( C ∞ ( Y p , Z p )) ∼ = ˇ H ∗ ( Y p , Z p ; C ) (2.171)We put all the above in a diagram HP ∗ ( I p − /I p ) ← HP ∗ ( O ( Y p , Z p )) → ˇ H ∗ ( Y p , Z p ; C ) ↓ ↓ (cid:107) HP ∗ ( J p − /J p ) ∼ = HP ∗ ( C ∞ ( Y p , Z p )) → ˇ H ∗ ( Y p , Z p ; C ) (2.172)The horizontal arrows are all natural isomorphisms, so the diagram commutes andthe vertical arrows are isomorphisms as well. (cid:50) K -theory and cyclic type homology theories hapter 3 Affine Hecke algebras
Here we commence our study of the main subjects of this thesis, affine Heckealgebras. They first appeared in the representation theory of certain topolog-ical groups, but that will be discussed only in the next chapter. Instead weconsider Hecke algebras as deformations of the group algebra of a Weyl group.More precisely, Iwahori-Hecke algebras are deformations of Coxeter groups, whileaffine Hecke algebras are deformations of affine Weyl groups. This deformation isachieved as follows. Let s be a typical generator of a Coxeter group W . There isa relation ( T s − T s + 1) = 0in Z [ W ]. We replace this relation by( T s − q ( s ))( T s + 1) = 0where the label q ( s ) can be any element of a commutative ring. In general itis possible to have different labels for different generators. Section 3.1 is mainlydedicated to making this precise, by providing the definitions of root data, labelfunctions and related objects.The affine Hecke algebra associated with these data will be denoted by H ( R , q ).If the labels are all positive then one can complete this to a C ∗ -algebra C ∗ r ( R , q )or, more subtly, to a Schwartz algebra S ( R , q ).We have two main goals in this chapter. On one hand we want to prepareeverything for a careful study of deformations in the parameters q , which we willundertake in Chapter 5. This dictates that we should provide explicit formulaswhenever possible.On the other hand we would like to apply the ideas developed in Chapter 2to affine Hecke algebras. Therefore it is imperative that we get a clear pictureof representations and the spectrum of H ( R , q ). This is provided by the work ofOpdam [98] on the Plancherel measure and the Fourier transform for affine Heckealgebras. It turns out that Prim( H ( R , q )) is a non-separated variety lying overa complex torus modulo a finite group, see Theorems 3.24 and 3.25. Similarly612 Chapter 3. Affine Hecke algebrasPrim( S ( R , q )) is a non-Hausdorff orbifold. On these spaces H ( R , q ) is relatedto polynomial functions, S ( R , q ) to smooth functions and C ∗ r ( R , q ) to continuousfunctions. In fact S ( R , q ) is isomorphic to an algebra of the type that we studiedin Section 2.5However, this is not enough, we also need the Langlands classification for H ( R , q ). In Section 3.2 we explain that it says essentially that Prim( S ( R , q ))is a deformation retract of Prim( H ( R , q )). The final form in which we will ac-tually apply this is the rather technical parametrization of irreducible H ( R , q )-representations Theorem 3.31. With all these preparations, and the interpretationof periodic cyclic homology as a cohomology theory on primitive ideal spectra,we can prove the main theorem of this chapter. It says that there are naturalisomorphisms HP ∗ ( H ( R , q )) ∼ = HP ∗ ( S ( R , q )) ∼ = K ∗ ( C ∗ r ( R , q )) ⊗ C (3.1) We give precise definitions of (most of) the objects needed to construct Heckealgebras. We do this both for Iwahori-Hecke algebras associated to Coxeter groupsand for affine Hecke algebras associated to root data.A Coxeter system (
W, S ) consists of a finite set S such that W is the groupgenerated by S , subject only to the relations( s i s j ) m ij = e for certain m ij ∈ { , , . . . , ∞} such that • m ij = 1 if and only if s i = s j • m ji = m ij Because the most important examples are Weyl groups, we denote the Coxetergroup by W and call the elements of S simple reflections. This simple definitionstill imposes a lot of structure, and indeed Coxeter groups have been studieddeeply. The most relevant results for us can be found for example in [61].A Coxeter system is completely determined by its Coxeter graph. This is agraph whose vertices correspond to elements of S . There is an edge between s i and s j if and only if m ij ≥
3, and it is labelled by this number m ij . A Coxetersystem is called irreducible if its Coxeter graph is connected.Just as for any finitely generated group, there is a natural length function (cid:96) on W , which assigns length 1 to any s ∈ S . To define it, pick w ∈ W and write it as w = s · · · s r If r ≥ w and (cid:96) ( w ) = r . Notice that (cid:96) ( w − ) = (cid:96) ( w ) since all the simple reflections have order 2..1. Definitions of Hecke algebras 63Depending on the numbers m ij , ( W, (cid:96) ) can be finite, of polynomial growth or ofexponential growth.If P ⊂ S then W P := (cid:104) P (cid:105) is a very special kind of subgroup of W , a standardparabolic subgroup. In general a parabolic subgroup of W is conjugate to some W P . The pair ( W P , P ) is a Coxeter system in its own right, and its length functionagrees with the restriction of (cid:96) to W P . Every right coset wW P contains a uniqueelement of minimal length, so there is a canonical set of representatives W P for W/W P . Moreover, if { P j } j are the connected components of the Coxeter graph of( W, S ) then W = (cid:76) j W P j . Hence one can learn a lot about Coxeter systems bystudying only irreducible ones.Let q : W → k be a function from W to a commutative unital ring k which islength-multiplicative, i.e. q ( wv ) = q ( w ) q ( v ) if (cid:96) ( wv ) = (cid:96) ( w ) + (cid:96) ( v ) (3.2)This is equivalent to giving a map q : S → k such that q ( s i ) = q ( s j ) whenever m ij is odd.We say that q is an equal label function in the special case that q ( s i ) = q ( s j )for all s i , s j ∈ S . In this situation it is customary to denote q ( s i ) simply by q , sothat q ( w ) = q (cid:96) ( w ) The Iwahori-Hecke algebra H ( W, q ) = H k ( W, q ) associated to the Coxeter group W ( S is usually suppressed from the notation) and the label function q is anassociative k -algebra which is a free k -module with bases { T w : w ∈ W } andfulfills the multiplication rules T w T v = T wv if (cid:96) ( wv ) = (cid:96) ( w ) + (cid:96) ( v ) (3.3) T s T s = ( q ( s ) − T s + q ( s ) T e if s ∈ S (3.4)It is proved in [61, Section 7.1] that such an object exists and is uniquely deter-mined by these conditions.Notice that if q ( s ) = 1 ∀ s ∈ S , then H ( W, q ) = k [ W ] is the group algebra of W over k . Any standard parabolic subgroup V gives rise to a parabolic subalgebra H (cid:0) V, q (cid:12)(cid:12) V (cid:1) , which as an A -module has bases { T v : v ∈ V } .Two choices of k are especially important. The first is simply k = C . Forthe second, write q i = q ( s i ) if s i ∈ S . Let q / i be indeterminates satisfying (cid:16) q / i (cid:17) = q ( s i ), and put k = Z (cid:104)(cid:8) q / i , q − / i (cid:9) s i ∈ S (cid:105) (3.5)In this case we have T − s = q ( s ) − T s + ( q ( s ) − − T e s ∈ S (3.6)4 Chapter 3. Affine Hecke algebrasSo T w is invertible in H ( W, q ) for any w ∈ W . Indeed, if w = s · · · s r is a reducedexpression, then T − w = T − s r · · · T − s To explain the introduction of the square roots we write N w = q ( w ) − / T w (3.7)These elements form again a bases of H ( W, q ), while (3.3) and (3.4) become some-what more manageable: N w N v = N wv if (cid:96) ( wv ) = (cid:96) ( w ) + (cid:96) ( v ) (3.8) (cid:0) N s i − q / i (cid:1)(cid:0) N s i + q − / i (cid:1) = 0 if s i ∈ S (3.9)We generalize the notion of an Iwahori-Hecke algebra as follows. Let Ω be agroup acting on W , and consider the semidirect product W (cid:111) Ω. Assume that ∀ w ∈ W, ω ∈ Ω (cid:96) ( ωwω − ) = (cid:96) ( w ) and q ( ωwω − ) = q ( w )so that we can extend (cid:96) and q to W (cid:111) Ω by (cid:96) ( wω ) := (cid:96) ( w ) and q ( wω ) := q ( w )In this situation the rules (3.3) and (3.4) again define a unique associative k -algebrawith basis { T g : g ∈ W (cid:111) Ω } . Such algebras are called extended Iwahori-Heckealgebras. Note that this is extremely general, since Ω can be nearly any group. Ifwe do not want to get too far away from proper Iwahori-Hecke algebras we haveto impose some restrictions on this group.We do this in the setting of an important class of such algebras, namely affineHecke algebras. We will mainly follow the notation of [98], which implies thatsometimes we attach a subscript 0 to a finite object, to distinguish it from itsaffine counterpart.First we introduce root data, for which we need the following objects. • X and Y are free abelian groups of the same finite rank, and (cid:104)· , ·(cid:105) : X × Y → Z is a perfect pairing between them • R ⊂ X and R ∨ ⊂ Y are finite subsets with a given bijection α → α ∨ The elements of R are called roots and the elements of R ∨ are called coroots.Define endomorphisms s α of X and s ∨ α of Y by s α ( x ) = x − (cid:104) x , α ∨ (cid:105) α (3.10) s ∨ α ( y ) = y − (cid:104) α , y (cid:105) α ∨ (3.11)For every α ∈ R we impose the conditions • (cid:104) α , α ∨ (cid:105) = 2.1. Definitions of Hecke algebras 65 • s α (cid:0) R (cid:1) = R • s ∨ α (cid:0) R ∨ (cid:1) = R ∨ A quadruple R = ( X, Y, R , R ∨ ) with these properties is a root datum. Further-more it is • reduced if Z α ∩ R = { α, − α } ∀ α ∈ R • semisimple if R ⊥ = { } ⊂ Y Let Q := Z R ⊂ X and Q ∨ := Z R ∨ ⊂ Y be the root lattice and the coroot lattice.The weight lattice is Hom Z ( Q ∨ , Z ) ⊃ Q . If R is semisimple then it contains X .These lattices do not necessarily span t ∗ := X ⊗ R and its linear dual t := Y ⊗ R ,but except for that R and R ∨ are root systems in the classical sense, and theyare dual to each other.Recall that a basis of R is a linearly independent subset F such that every α ∈ R can be written as α = (cid:88) β ∈ F n β β where either n β ∈ Z ≥ ∀ β ∈ F or n β ∈ Z ≤ ∀ β ∈ F This gives a partition R = R +0 ∪ R − . We call the roots in F simple, those in R +0 positive and those in R − negative. Bases always exist, and we will assumethat one is given with the root datum, which we will henceforth write as R = ( X, Y, R , R ∨ , F ) (3.12)There are several ways to construct new root data from a given one. Firstly, wecan take the direct product of two root data: R × R (cid:48) := ( X × X (cid:48) , Y × Y (cid:48) , R ∪ R (cid:48) , R ∨ ∪ R (cid:48) ∨ , F ∪ F (cid:48) ) (3.13)In particular we always have the ”trivial one dimensional extension” R × Z := ( X × Z , Y × Z , R , R ∨ , F ) (3.14)Alternatively one can simply exchange the roles of X and Y . Then R ∨ = ( Y, X, R ∨ , R , F ∨ ) (3.15)is the dual root datum of R . Furthermore, for P ⊂ F write R P = Q P ∩ R and R ∨ P = Q P ∨ ∩ R ∨ This R P is a parabolic root subsystem of R , and with it we associate the rootdatum R P := (cid:0) X, Y, R P , R ∨ P , P (cid:1) (3.16)6 Chapter 3. Affine Hecke algebrasFinally, define X P = X (cid:14)(cid:0) X ∩ ( P ∨ ) ⊥ (cid:1) Y P = Y ∩ Q P ∨ X P = X/ ( X ∩ Q P ) Y P = Y ∩ P ⊥ R P = (cid:0) X P , Y P , R P , R ∨ P , P (cid:1) (3.17)Let us have a look at the various Weyl groups associated to R . Clearly every s α induces a reflection of t , and they generate a finite Coxeter group W , the Weylgroup of R . In accordance with the terminology for roots, we can take S = { s α : α ∈ F } (3.18)as the set of simple reflections. The action of W on X is by group homomorphisms,so we can construct the semidirect product W = W ( R ) := X (cid:111) W (3.19)By identifying x ∈ X with the translation t x , we can regard W as a group of affinelinear transformations of X . This W is the Weyl group of R , and in the same waywe construct its normal subgroup W aff := Q (cid:111) W (3.20)which we call the affine Weyl group of either R or R . Although W may beisomorphic to Z n (if R = ∅ ) , W aff is always a Coxeter group, and it is possible toextend S to a set of Coxeter generators for W aff . To do so, observe that pairingwith F defines a partial ordering on Y , and let F ∨ m be the set of maximal elementsof R ∨ for this ordering. For every α ∨ ∈ F ∨ m we consider the affine reflection t α s α : x → x − (cid:104) x , α ∨ (cid:105) α + α (3.21)in the hyperplane { x ∈ X : (cid:104) x , α ∨ (cid:105) = 1 } . The desired set of generators is S aff := S ∪ { t α s α : α ∨ ∈ F ∨ m } (3.22)The Coxeter graph of R is that of ( W aff , S aff ). It is obtained from the Coxetergraph of (cid:0) W , S (cid:1) by suitably adding exactly one vertex to every connected com-ponent. Hence a standard parabolic subgroup W P ⊂ W aff is infinite whenever | P | > | F | . If ( W aff , S aff ) is irreducible, then any proper parabolic subgroup of W aff is finite.We have a length function (cid:96) on W aff , which however does not immediatelyextend to W . To achieve that we need a different characterization of (cid:96) . It is wellknown that in the finite Weyl group W the length of w is the number of positiveroots made negative by w : (cid:96) ( w ) = (cid:8) α ∈ R +0 : w ( α ) ∈ R − (cid:9) = (cid:12)(cid:12) R +0 ∩ w − (cid:0) R − (cid:1)(cid:12)(cid:12) (3.23)The same holds in W aff , one only has to replace R by an affine root system. [65,Propostion 1.23] says.1. Definitions of Hecke algebras 67 Proposition 3.1
The following formula defines a natural extension of (cid:96) to W : (cid:96) ( wt x ) = (cid:88) α ∈ R +0 ∩ w − ( R − ) |(cid:104) x , α ∨ (cid:105) +1 | + (cid:88) α ∈ R +0 ∩ w − ( R +0 ) |(cid:104) x , α ∨ (cid:105)| w ∈ W , x ∈ X Moreover the set
Ω := { ω ∈ W : (cid:96) ( ω ) = 0 } is a subgroup of W , complementary to W aff : W = W aff (cid:111) ΩHence we can say that a reduced expression for w ∈ W is a decomposition w = vω or w = ωv , where ω ∈ Ω and v ∈ W aff is written in a reduced way. Define X + := { x ∈ X : (cid:104) x , α ∨ (cid:105) ≥ ∀ α ∈ F } X − := { x ∈ X : (cid:104) x , α ∨ (cid:105) ≤ ∀ α ∈ F } = − X + (3.24)It is well known that Z ( W ) = X + ∩ X − W X + = X (3.25)It follows immediately from Proposition 3.1 that for x ∈ X + (cid:96) ( t x ) = (cid:88) α ∈ R +0 (cid:104) x , α ∨ (cid:105) (3.26)so (cid:96) is additive on the abelian semigroup X + ⊂ W .Our definition of a label function is more restrictive than that for an extendedIwahori-Hecke algebra. It is a function q : W → C × such that ∀ w, v ∈ W, ω ∈ Ω • q ( ω ) = 1 • q ( wv ) = q ( w ) q ( v ) if (cid:96) ( wv ) = (cid:96) ( w ) + (cid:96) ( v )The set of label functions is in bijection with the set of functions q : S aff → C × such that q ( s i ) = q ( s j ) whenever s i and s j are conjugate in W .One can also describe q in terms of a function on a root system associated to R . Assume for simplicity that R is reduced, and define a non-reduced root sytemin X : R nr := R ∪ { α : α ∨ ∈ Y } (3.27)Obviously we write (2 α ) ∨ = α ∨ /
2, and we let R be the root system of long rootsin R nr : R := { α ∈ R nr : α ∨ / ∈ Y } (3.28)The set of simple long roots is F := { α ∈ R : α ∈ F or α/ ∈ F } (3.29)8 Chapter 3. Affine Hecke algebrasFor α ∈ R \ R and β ∈ R ∩ R we put q α ∨ = q ( t α s α ) q α ∨ / = q ( s α ) q ( t α s α ) − (3.30) q β ∨ = q ( s β ) = q ( t β s β ) q β ∨ / = q β ∨ = 1 (3.31)Since β ∨ / / ∈ Y and 2 β ∨ / ∈ R ∨ nr , we can ignore them here. However we includethese conventions because they will simplify some future notations. Clearly q : R ∨ nr → C × is W -invariant, and conversely every W -invariant function R ∨ nr → C × determines a unique label function W → C × as above. This correspondenceimplies the following formulas [97, Corollaries 1.3 and 1.5] Corollary 3.2
For w ∈ W and x ∈ X + q ( w ) = (cid:89) α ∈ R + nr ∩ w − ( R − nr ) q α ∨ and q ( t x ) = (cid:89) α ∈ R + nr q (cid:104) x , α ∨ (cid:105) α ∨ Given a reduced root datum R = ( X, Y, R , R ∨ , F ) and a label function q , wedefine the affine Hecke algebra H ( R , q ) as the unique associative C -algebra whichhas basis { T w : w ∈ W } and satisfies the multiplication rules T w T v = T wv if (cid:96) ( wv ) = (cid:96) ( w ) + (cid:96) ( v ) T s T s = ( q ( s ) − T s + q ( s ) T e if s ∈ S aff (3.32)This algebra is canonically isomorphic to the crossed product of an Iwahori-Heckealgebra and the group of elements of length 0 in W : H ( R , q ) ∼ = H ( W aff , q ) (cid:111) Ω (3.33)Our affine Hecke algebra has a large commutative subalgebra A , isomorphic to thegroup algebra of X . We will regard this also as O ( T ), where T = Hom Z (cid:0) X, C × (cid:1) ∼ = Prim (cid:0) C [ X ] (cid:1) (3.34)The action of W on X induces an action on T by( w · t )( x ) = t ( w − x ) (3.35)To identify this algebra A , let q / : W → C be a label function such that q / ( w ) = q ( w ) ∀ w ∈ W . Abbreviate q / ( w ) − = q − / ( w ) and q / ( s i ) − q − / ( s i ) = η i (3.36)In terms of the new bases (cid:110) N w = q − / ( w ) T w : w ∈ W (cid:111) the multiplication rules for H ( R , q ) become N w N v = N wv if (cid:96) ( wv ) = (cid:96) ( w ) + (cid:96) ( v ) (cid:0) N s − q / ( s ) (cid:1)(cid:0) N s + q − / ( s ) (cid:1) = 0 if s ∈ S aff (3.37).1. Definitions of Hecke algebras 69Furthermore every N w is invertible, and N − s i = N s i − η i N e s i ∈ S aff (3.38)Because (cid:96) is additive on X + , the map X + → H ( R , q ) × : x → N t x (3.39)is a monomorphism of semigroups. Extend it to a group homomorphism θ : X →H ( R , q ) × by defining θ x = N t y N − t z = N − t z N t y (3.40)if x = y − z with y, z ∈ X + . This independent of the choice of y and z .The following theorem is due to Bernstein, Lusztig and Zelevinski, see [86, Sec-tion 3]. It describes what is also known as the Bernstein presentation of H ( R , q ). Theorem 3.3
1. The sets { N w θ x : w ∈ W , x ∈ X } and { θ x N w : w ∈ W , x ∈ X } are both bases of H ( R , q ) .2. The subalgebra A := span { θ x : x ∈ X } is naturally isomorphic to C [ X ] .The Weyl group W acts on A by w · θ x = θ wx , or equivalently ( w · a )( t ) = a w ( t ) := a ( w − t ) t ∈ T, a ∈ A ∼ = O ( T )
3. The center of H ( R , q ) is Z ( H ( R , q )) = A W ∼ = O ( T ) W ∼ = O ( T /W )
4. Take a ∈ A , α i ∈ F and let s be the unique vertex of the Coxeter graph of R which is connected to s i = s α i but does not lie in S . Then aN s i − N s i a s i = (cid:26) η i ( a − a s i )( θ − θ − α i ) − if α ∨ i / ∈ Y ( η i + η θ − α i )( a − a s i )( θ − θ − α i ) − if α ∨ i ∈ Y (3.41)Equations (3.41) are also known as the Bernstein-Lusztig-Zelevinski relations.Since A is of finite rank over A W , it follows that H ( R , q ) is of finite rank as amodule over its center. In particular it is a finite type algebra in the sense ofSection 2.2.For P ⊂ F we can use the above to define label functions q P and q P onthe root data R P and R P . For q P , use (3.30) and (3.31) as a definition forall α, β ∈ R P , and extend this to q P : W (cid:0) R P (cid:1) → C × . Notice that q P ( t x ) = 1whenever x ⊥ P ∨ . Now q P : W ( R P ) → C × is simply the map induced by q P . Theaffine Hecke algebra H (cid:0) R P , q P (cid:1) can be identified with the parabolic subalgebra of H ( R , q ) generated by A and H (cid:0) W P , q (cid:12)(cid:12) W P (cid:1) . Furthermore H ( R P , q P ) is naturallya quotient of H (cid:0) R P , q P (cid:1) .Notice also that if R = R × R , then q restricts to label functions on the Weylgroups associated with R and R , and there is a canonical identification H ( R , q ) ∼ = H ( R , q ) ⊗ H ( R , q ) (3.42)0 Chapter 3. Affine Hecke algebrasTo study the above algebras we introduce the complex tori T P = Hom Z ( X P , C × ) = (cid:8) t ∈ T : t ( x ) = 1 if x ⊥ P ∨ (cid:9) T P = Hom Z (cid:0) X P , C × (cid:1) = (cid:8) t ∈ T : t ( x ) = 1 if x ∈ Q P (cid:9) (3.43)They decompose into a unitary and a real split part: T P = T P,u × T P,rs = Hom Z (cid:0) X P , S (cid:1) × Hom Z ( X P , R + ) T P = T Pu × T Prs = Hom Z (cid:0) X P , S (cid:1) × Hom Z (cid:0) X P , R + (cid:1) (3.44)Notice that K P = T P ∩ T P = T Pu ∩ T P,u is a finite group, not necessarily equal to { } . We make the identificationsLie (cid:0) T Prs (cid:1) = t P = Y P ⊗ Z R Lie (cid:0) T P (cid:1) = t P ⊗ R C = Y P ⊗ Z C Lie (cid:0) T Pu (cid:1) = i t P = Y P ⊗ Z i R (3.45)Also define the positive parts t P, + = (cid:8) λ ∈ t P : (cid:104) α , λ (cid:105) > ∀ α ∈ F \ P (cid:9) T P, + rs = (cid:8) t ∈ T Prs : t ( α ) > ∀ α ∈ F \ P (cid:9) = exp (cid:0) t P, + (cid:1) (3.46) This section is meant as an introduction to the representation theory of affineHecke algebras. None of the results presented here are original, varying fromclassical (Theorem 3.15) to very recent (Proposition 3.10).Since an affine Hecke algebra is of finite type over its center, all its irre-ducible representations have finite dimension. Therefore we mainly study fi-nite dimensional representations. We give two partial classifications of H ( R , q )-representations. One is in terms of their restrictions to subalgebras associatedwith finite Coxeter groups. The other is more important, and in the spirit ofLanglands. It shows that the study of H ( R , q )-representations can be reduced tothe study of so-called tempered representations. After that we define the C ∗ andSchwartz completions C ∗ r ( R , q ) and S ( R , q ) of an affine Hecke algebra. It turnsout that S ( R , q )-representations are characterized among H ( R , q )-representationsby the requirement that they are tempered.Let H = H ( R , q ) be an affine Hecke algebra and Rep( H ) its category of finitedimensional representations. Since A ∼ = C [ X ] is commutative, every irreducible.2. Representation theory 71 A -module is onedimensional, of the form C t for a character t ∈ T . For ( π, V ) ∈ Rep( H ) and t ∈ T we put V t = { v ∈ V : ∃ n ∈ N : ( π ( a ) − a ( t )) n v = 0 ∀ a ∈ A} (3.47)If V t (cid:54) = 0 then there exists an eigenvector v ∈ V with π ( a ) v = a ( t ) v ∀ a ∈ A . Inthis case t is called an A -weight of V , and V t a generalized weight space. As an A -module V is the direct sum of the nonzero V t .Let us consider principal series representations. By definition they are therepresentations I t = Ind HA ( C t ) for t ∈ T (3.48)They have dimension | W | , and we realize them all on the vector space H ( W , q ) ∼ = H / (cid:104){ a − a ( t ) : a ∈ A}(cid:105) (3.49)The importance of principal series representations is already clear from the follow-ing well-known result. Lemma 3.4
1. Every irreducible H -representation ( π, V ) is a quotient of some I t
2. If I t ( h ) = 0 for all t in a Zariski-dense subset of T , then h = 0
3. Jac ( H ) = 0 Proof. A -weight t of V and a correspondingeigenvector v ∈ V . Define an H -module homomorphism I t → V : h → π ( h ) v (3.50)This is surjective because V is irreducible.2 is based upon Theorem 3.3. The function T → End C ( H ( W , q )) : t → I t ( h ) (3.51)is polynomial, and since it is 0 on a Zariski-dense subset it vanishes identically.Write h = (cid:80) w ∈ W a w T w with a w ∈ A and suppose that h (cid:54) = 0. Then we can find w (cid:48) ∈ W such that a w (cid:48) (cid:54) = 0 and (cid:96) ( w (cid:48) ) is maximal with respect to this property.From (3.41) we see that I t ( h )( T e ) = (cid:88) w ∈ W b w ( t ) T w with b w (cid:48) = a w (cid:48) There (3.51) is not identically 0, so our assumption h (cid:54) = 0 must be false.3. By [90, (3.4.5)] or [71, Theorem 2.2] there is a nonempty Zariski-open subset T (cid:48) of T such that I t is irreducible ∀ t ∈ T (cid:48) . So if h ∈ Jac( H ) then h = 0 by part 2. (cid:50) Z -weights of V , where Z = Z (cid:0) H ( R , q ) (cid:1) is the center of H . Using 3.3.2 we identify Prim( Z ) with T /W . Assume that Z acts by scalars on V , which by Schur’s lemma is the case if V is irreducible. Thenthe central character of V is the unique orbit W t ∈ T /W such that π ( a ) v = a ( t ) v ∀ v ∈ V, a ∈ Z
For any W -stable U ⊂ T let Rep U ( H ( R , q )) be the category of all finite dimen-sional H -modules whose Z -weights are contained in U/W . There are a few waysto ”localize” the algebra H at U , i.e. to construct an algebra whose representationsare precisely Rep U ( H ( R , q )). One way, suitable for open U , is by tensoring (over A ) with analytic functions on U , as we shall see in (3.116). Another way, whichworks best if U is a closed subvariety of T , is by completing H with respect to theideal J U = { h ∈ H : I t ( h ) = 0 ∀ t ∈ U } Notice that J U is generated by { a ∈ Z : a (cid:12)(cid:12) U = 0 } . Explicitly, we get the modules V U = V ⊗ H H U H U = lim n ←∞ H /J nU (3.52)For U = W t with t ∈ T this was done in [86, Section 7], and it is consistent with(3.47) in the sense that V W t = (cid:88) w ∈ W V wt (3.53)Finally, we can vary on this construction in a less subtle fashion, by replacing H U in (3.52) with H /J U . This has the advantage that we reduce things to modulesover finite dimensional algebras.Now we start the preparations for the Langlands classification, which can befound in [40]. We say that V is a tempered H -module if | x ( t ) | ≤ x ∈ X + and every A -weight of V . The explanation will follow in Lemma 3.14.Contrarily, we call V anti-tempered if | x ( t ) | ≥ x and t . And lessrestrictively we say that V is essentially tempered if | x ( t ) | ≤ A -weight t of V and every x ∈ Z R ∩ X + . The only difference with tempered is that t (cid:12)(cid:12) Z ( W ) need not be a unitary character. Lemma 3.5
Let ( π, V ) be an essentially tempered H -representation which admitsa central character W rt , with t ∈ T F rs and | r | ∈ T F ,rs . There exists an automor-phism ψ t of H such that ( π ◦ ψ − t , V ) is a tempered H -representation with centralcharacter W r .Proof. Define ψ t ( N w θ x ) = t ( x ) N w θ x (3.54).2. Representation theory 73This is an automorphism because t is 1 on Z R . Let t , . . . , t d be the A -weightsof ( π, V ). Clearly, the A -weights of ( π ◦ ψ − t , V ) are t t − , . . . , t d t − . For x ∈ Z R ∩ X + we have | t i t − ( x ) | = | t i ( x ) | ≤ π is essentially tempered, and for x ∈ X + ∩ X − we have | t i t − ( x ) | = 1because | t i | ∈ W | r | ⊂ T F ,rs . Hence ( π ◦ ψ − t , V ) is tempered with central char-acter W t i t − = W r. (cid:50) In general, let ( σ, V σ ) be any finite dimensional representation of H P = H ( R P , q P ). Recall [61, Proposition 1.10] that W P = { w ∈ W : w ( P ) ⊂ R +0 } (3.55)is the set of minimal length representatives of W /W P . Construct the vector space H (cid:0) W P (cid:1) = span (cid:8) N w : w ∈ W P (cid:9) ⊂ H ( W , q ) (3.56)The H -representation π ( P, σ ) := Ind HH P ( σ ) (3.57)can be realized on H (cid:0) W P (cid:1) ⊗ C V σ . From the proof of [98, Proposition 4.20] we cansee what the weights of this representation are: Lemma 3.6
The A -weights of π ( P, σ ) are precisely the elements w ( t ) , where t isan A -weight of ( σ, V σ ) and w ∈ W P . If σ is irreducible then it has a central character W P r ∈ T /W P . Since T rs = T Prs × T P,rs (3.58)and W P acts trivially on T Prs , there is a unique r σ ∈ T Prs such that W P | r | = W P r (cid:48) r σ for some r (cid:48) ∈ T P,rs
Let Λ be the set of all pairs (
P, σ ), where σ is an irreducible essentially temperedrepresentation of H P . The set of Langlands data isΛ + = (cid:8) ( P, σ ) ∈ Λ : r σ ∈ T P, + rs (cid:9) (3.59)Now we can state the Langlands classification for affine Hecke algebras: Theorem 3.7
Let ( P, σ ) ∈ Λ + . The H -module π ( P, σ ) is indecomposable and hasa unique irreducible quotient L ( P, σ ) . For every irreducible H -representation π there is precisely one Langlands datum ( P, σ ) ∈ Λ + such that π is equivalent to L ( P, σ ) . Proof.
This is entirely analogous to the corresponding statements for graded Heckealgebras, which were proved by Evens [45, Theorem 2.1]. See also [40, Section 6]. (cid:50)
Another way to study H -representations is by their restrictions to simplersubalgebras. We do this by constructing a nice (projective) resolution of an H -module, which stems from joint work of Opdam and Reeder, cf. [99, Section 8].Number the s i ∈ S aff such that the elements corresponding to one connectedcomponent of the Coxeter graph of ( W aff , S aff ) are numbered successively. For I ⊂ { , , . . . , | S aff |} let C [ I ] be the vector space with basis { e i : i ∈ I } and W I the standard parabolic subgroup of W aff generated by { s i : i ∈ I } . Recall thatthe ”length 0” subgroup Ω of W acts on S aff by conjugation. Transfer this to anaction of Ω on the indices i and putΩ I := { ω ∈ Ω : ω ( I ) = I } By definition Ω I acts on W I , so the extended Iwahori-Hecke algebra H ( R , I, q ) = H ( W I , q ) (cid:111) Ω I is well-defined. Note that W I can be either finite or infinite, but that we alwayshave X + ∩ X − = Z ( W ) ⊂ Ω I . If R is semisimple and W I is finite, then H ( R , I, q )has finite dimension.Because we want to define a conjugate-linear, anti-multiplicative involution on H ( R , q ), from now we will we assume the following. Condition 3.8
The label function of an affine Hecke algebra only takes values in (0 , ∞ ) Lemma 3.9
Suppose that W I is finite and let χ be a character of Z ( W ) . Then H ( R , I, q ) χ := H ( R , I, q ) / ker (cid:16) Ind H ( R ,I,q ) C [ Z ( W )] C χ (cid:17) is a finite dimensional semisimple algebra.Proof. As vector spaces we may identify H ( R , I, q ) χ = Ind H ( R ,I,q ) C [ Z ( W )] C χ = H ( W I , q ) ⊗ C [Ω I /Z ( W )]We can extend | χ | canonically to X ⊗ R , making it 1 on R . Using this extensionwe define an involution ∗ χ on H ( R , I, q ) by( h w T w ) ∗ χ = h w | χ | (2 w (0)) T w − This map is antimultiplicative by Condition 3.8. The associated bilinear form is (cid:104) h , h (cid:48) (cid:105) χ = x e if h ∗ χ · h (cid:48) = (cid:88) w ∈ W x w N w .2. Representation theory 75By construction Ind H ( R ,I,q ) C [ Z ( W )] C χ is now a unitary representation. This makes H ( R , I, q ) χ into a finite dimensional Hilbert algebra, so in particular it issemisimple. (cid:50) For ( π, V ) ∈ Rep( H ) and n ∈ N consider the H -module P n ( V ) = (cid:77) | I | = n, | W I | < ∞ H ⊗ H ( W I ,q ) (cid:111) Z ( W ) V (cid:12)(cid:12) H ( W I ,q ) (cid:111) Z ( W ) ⊗ C n (cid:94) C [ I ] (3.60)Here (cid:111) Z ( W ) is just an abbreviaton of ⊗ C [ Z ( W )]. We put P | F | +1 ( V ) = V and P n ( V ) = 0 if n < n ≥ | F | + 2. Define H -module homomorphisms d n : P n ( V ) → P n +1 ( V ) d n ( h ⊗ H ( W I ,q ) (cid:111) Z ( W ) v ⊗ λ ) = (cid:77) j : | W I ∪{ j } | < ∞ h ⊗ H ( W I ∪{ j } ,q ) (cid:111) Z ( W ) v ⊗ λ ∧ e j (3.61)Notice that actually the sum runs only over j / ∈ I , for otherwise λ ∧ e j = 0. Toconstruct a fitting map d | F | we need to exert ourselves a little more. We introducea sign function bysign( e n ∧ · · · ∧ e n k ) = (cid:26) e n ∧ · · · ∧ e n k ∧ η = e ∧ e ∧ · · · ∧ e | S aff | − e n ∧ · · · ∧ e n k ∧ η = − e ∧ e ∧ · · · ∧ e | S aff | where η is the wedge product of the e i with 1 ≤ i ≤ | S aff | and i (cid:54) = n j , in standardorder. Using this convention we define d | F | : P | F | ( V ) → Vd | F | ( h ⊗ H ( W I ,q ) (cid:111) Z ( W ) v ⊗ λ ) = sign( λ ) π ( h ) v Looking at the ∧ -terms we see that d n +1 ◦ d n = 0, so ( P ∗ ( V ) , d ∗ ) is an augmenteddifferential complex. The group Ω acts naturally on this complex by ω ( h ⊗ H ( W I ,q ) (cid:111) Z ( W ) v ⊗ λ ) = hT − ω ⊗ H ( W ω ( I ) ,q ) (cid:111) Z ( W ) π ( T ω ) v ⊗ ω ( λ ) (3.62)This action commutes with the action of H and with the differentials d n , so (cid:0) P ∗ ( V ) Ω , d ∗ (cid:1) is again a differential complex. Proposition 3.10 −→ P ( V ) Ω d −−→ P ( V ) Ω d −−→ · · · −→ P | F | ( V ) Ω d | F | −−−→ V → is a natural resolution of V by finitely generated modules. If V admits the Z ( W ) -character χ then every module P n ( V ) Ω is projective in the category of all H -modules with Z ( W ) -character χ . Proof.
This result a generalization of [99, Proposition 8.1] to root data that arenot semisimple. The proof is based upon [73, Section 1], as indicated by Opdamand Reeder.First we consider the case Ω = Z ( W ) = { e } , W = W aff . There is a linearbijection φ : C [ W ] ⊗ C V → H ⊗ C Vφ ( w ⊗ v ) = T w ⊗ π ( T w ) − v (3.63)For s i ∈ S aff we write L i := span { hT s i ⊗ π ( T s i ) − v − h ⊗ v : h ∈ H , v ∈ V } ⊂ H ⊗ C V C [ W ] i := (cid:8) (cid:80) w ∈ W x w w : x ws i = − x w ∀ w ∈ W (cid:9) ⊂ C [ W ](3.64)This L i is interesting because H ⊗ H ( W I ,q ) V = (cid:0) H ⊗ C V (cid:1)(cid:46) (cid:88) i ∈ I L i (3.65)Let w ∈ W be such that (cid:96) ( ws i ) > (cid:96) ( w ). φ (( ws i − w ) ⊗ v ) = T ws i ⊗ π ( T ws i ) − v − T w ⊗ π ( T w ) − v = T w T s i ⊗ π ( T s i ) − π ( T w ) − v − T w ⊗ π ( T w ) − v ∈ L i (3.66)so φ ( C [ W ] i ⊗ V ) ⊂ L i . On the other hand, L i is spanned by elements as in (3.64)with h = T w or h = T ws i . φ − ( T ws i T s i ⊗ π ( T s i ) − v − T ws i ⊗ v ) = φ − ( q i T w + ( q i − T ws i ⊗ π ( T s i ) − v ) − ws i ⊗ π ( T ws i ) v = q i w ⊗ π ( T w ) π ( T s i ) − v + ( q i − ws i ⊗ π ( T ws i ) π ( T s i ) − v − ws i ⊗ π ( T ws i ) v = q i ( w − ws i ) ⊗ π ( T w T − s i ) v + ws i ⊗ π (cid:0) q i T w T − s i + ( q i − T ws i T − s i − T ws i (cid:1) v =( w − ws i ) ⊗ π ( T w q i T − s i ) v + ws i ⊗ π (cid:0) T w ( T s i + 1 − q i ) + ( q i − T w − T w T s i (cid:1) v =( w − ws i ) ⊗ π (cid:0) T w ( T s i + 1 − q i ) (cid:1) v ∈ C [ W ] i ⊗ V (3.67)We conclude that φ − ( L i ) = C [ W ] i ⊗ V . Now we bring the linear bijections C [ W ] (cid:46) (cid:88) i ∈ I C [ W ] i → C [ W/W I ] : w → wW I (3.68)into play. Under these identifications our differential complex becomes0 → C [ W ] ⊗ V ⊗ C → · · · → (cid:77) | I | = n, | W I | < ∞ C [ W/W I ] ⊗ V ⊗ n (cid:94) C [ I ] → · · · → V → d | F | is surjective it suffices to show that the complex C (cid:48) n = (cid:76) | I | = n, | W I | < ∞ C [ W/W I ] ⊗ (cid:86) n C [ I ] d (cid:48) n ( wW I ⊗ λ ) = (cid:80) j : | W I ∪{ j } | < ∞ wW I ∪{ j } ⊗ λ ∧ e j (3.69).2. Representation theory 77has cohomology H (cid:48) n = (cid:26) C if n = | F | n (cid:54) = | F | (3.70)This is best seen by a geometrical interpretation. It is well-known that the alcove C ∅ := (cid:8) x ∈ Q ⊗ Z R : (cid:104) x , α ∨ (cid:105) ≥ ∀ α ∨ ∈ F ∨ , (cid:104) x , β ∨ (cid:105) ≤ ∀ β ∨ ∈ F ∨ m (cid:9) is a fundamental domain for the action of W on Q ⊗ Z R . The finite groups W I are naturally identified with the stabilizers of the faces C I of C ∅ . Thus everycoset wW I corresponds to a polysimplex wC I of dimension | F | − | I | . It followsthat ( C (cid:48)∗ , d (cid:48)∗ ) is the complex that computes H | F |−∗ ( Q ⊗ Z R ; C ) by means of a(polysimplicial) triangulation. Together with the Poincar´e Lemma this leads to(3.70), proving the proposition in the special case Ω = { e } .Now the general case. Clearly P n ( V ) is finitely generated because V has finitedimension and because there are only finitely many I ’s involved.Since the action of Ω on S aff factors through a finite group we can construct aReynolds operator R Ω := [Ω : Z ( W )] − (cid:88) ω ∈ Ω /Z ( W ) ω ∈ End H (cid:0) P n ( V ) (cid:1) It follows that P n ( V ) Ω = R Ω · P n ( V ) is a direct summand of P n ( V ), and hencealso finitely generated.We generalize (3.63) to a bijection φ : C [ W/Z ( W )] ⊗ C V → H ⊗ C [ Z ( W )] Vφ ( w ⊗ v ) = T w ⊗ π ( T w ) − v (3.71)Just as above this leads to bijections (cid:77) | I | = n, | W I | < ∞ C [ W/ ( W I × Z ( W ))] ⊗ V ⊗ n (cid:94) C [ I ] → P n ( V ) (3.72)Since both sides are free Ω /Z ( W )-modules we see that (cid:77) | I | = n, | W I | < ∞ C [ W aff /W I ] ⊗ V ⊗ n (cid:94) C [ I ] ∼ = P n ( V ) Ω (3.73)Now the above geometrical argument shows that the P n ( V ) Ω do indeed form aresolution of V .Assume now that V admits a Z ( W )-character χ . Then π (cid:12)(cid:12) H ( R ,I,q ) factorsthrough H ( R , I, q ) χ and by Lemma 3.9 it is projective as a representation of thisalgebra. This implies that P n ( V ) and its direct summand P n ( V ) Ω are projectivein the category of H -representations with Z ( W )-character χ. (cid:50) Corollary 3.11
Suppose that
V, V (cid:48) ∈ Rep( H ) are such that V (cid:12)(cid:12) H ( R ,I,q ) and V (cid:48) (cid:12)(cid:12) H ( R ,I,q ) are equivalent whenever W I is finite. Then V and V (cid:48) define the sameclass in the Grothendieck group of finitely generated H -modules.Proof. By assumption we can find a collection of H ( R , I, q )-module isomorphisms α I : V → V (cid:48) such that π (cid:48) ( T ω ) α I = α ω ( I ) π ( T ω ) ∀ ω ∈ ΩThese combine to H -module isomorphisms α n : P n ( V ) Ω → P n ( V (cid:48) ) Ω α n ( h ⊗ H ( W I ,q ) (cid:111) Z ( W ) v ⊗ λ ) = h ⊗ H ( W I ,q ) (cid:111) Z ( W ) α I ( v ) ⊗ λ Hence by Proposition 3.10 V and V (cid:48) have equivalent finitely generated resolutions. (cid:50) Unfortunately this result is not very strong, as the class of V is nearly always0. This certainly is the case if V is of the form π ( P, σ ) with P (cid:54) = F .Let G ( H ) be the Grothendieck group of finite dimensional H -modules, and K ( H ) the Grothendieck group of finitely generated projective H -modules. TheEuler-Poincar´e pairing [112, Section III.4] on G ( H ) is defined as EP ( V, V (cid:48) ) = ∞ (cid:88) n =0 ( − n dim Ext n H ( V, V (cid:48) ) (3.74)where Ext n H is the higher derived functor of Hom H .Let us recall some relevant observations from [99, Section 8]. Assume that R issemisimple. By Proposition 3.10 the Euler characteristic can be defined for finitedimensional H -modules byEul : G ( H ) → K ( H )Eul [ V ] = | F | (cid:88) n =0 ( − | F |− n (cid:2) P n ( V ) Ω (cid:3) (3.75)Moreover there is a natural pairing between G ( H ) and K ( H ). Given a represen-tation π of H and an idempotent p ∈ M n ( C ) ⊗ H we put (cid:2) [ p ] , [ π ] (cid:3) = rank (id ⊗ π )( p ) ∈ Z (3.76)Because ( P ∗ ( V ) Ω , d ∗ ) is a projective resolution of V we have the equalities EP ( V, V (cid:48) ) = | F | (cid:88) n =0 ( − | F |− n dim Hom H (cid:0) P n ( V ) Ω , V (cid:48) (cid:1) = (cid:2) Eul [ V ] , [ V (cid:48) ] (cid:3) (3.77).2. Representation theory 79But the modules P n ( V ) are induced from semisimple subalgebras of finite dimen-sion, so this can be expressed more explicitly. By Frobenius reciprocity we havedim Hom H ( P n ( V ) , V (cid:48) ) = (cid:88) | I | = n, | W I | < ∞ dim Hom H ( W I ,q ) (cid:111) Z ( W ) ( V, V (cid:48) ) (3.78)Let (cid:15) I be the character of the Ω I -representation (cid:86) | I | C [ I ]. Taking the Ω-invariantsof P n ( V ) in (3.78) we find EP ( V, V (cid:48) ) = (cid:88) I ⊂ S aff , | W I | < ∞ ( − | F |−| I | [Ω : Ω I ] dim Hom H ( R ,I,q ) ( V ⊗ (cid:15) I , V (cid:48) ) (3.79)Thus the Euler-Poincar´e pairing of two finite dimensional H -modules is completelydetermined by their restrictions to the finite dimensional semisimple subalgebras H ( R , I, q ). Because such algebras have only finitely many inequivalent irreduciblemodules, this pairing is symmetric and invariant under continuous deformationsof its input. However, from (3.74) we quickly deduce that modules with differentcentral characters are orthogonal for EP . We will see in (3.92) that every moduleof the form π ( P, σ ) with P (cid:40) F admits continuous deformations of its centralcharacter. Therefore all such modules are in the radical of EP .So far for the purely algebraic properties of affine Hecke algebras, we are alsointerested in their analytic structure. Condition 3.8 gives us a canonical way todefine the label function q / and the basis elements N w . Let x = (cid:80) w ∈ W x w N w be an element of H ( R , q ) and define its adjoint and its trace by x ∗ := (cid:88) w ∈ W x w N w − and τ ( x ) = x e (3.80)Condition (3.8) assures that indeed τ is positive and that ( xy ) ∗ = y ∗ x ∗ . This leadsto a bitrace or Hermitian inner product (cid:104) x , y (cid:105) := τ ( x ∗ y ) x, y ∈ H ( R , q ) (3.81)and a norm (cid:107) x (cid:107) τ := (cid:112) (cid:104) x , x (cid:105) = (cid:112) τ ( x ∗ x ) (3.82)By a simple calculation one can show that { N w : w ∈ W } is an orthonormal basesof H ( R , q ) for this inner product. The bitrace (cid:104)· , ·(cid:105) gives H ( R , q ) the structureof a Hilbert algebra, in the sense of [41, Appendice A 54]. Let H ( R , q ) be itsHilbert space completion, and B ( H ( R , q )) the associated C ∗ -algebra of boundedoperators. Consider the multiplication maps λ ( x ) : y → xyρ ( x ) : y → yx H ( R , q )of the same norm, which we denote by (cid:107) x (cid:107) o := (cid:107) λ ( x ) (cid:107) B ( H ( R ,q )) = (cid:107) ρ ( x ) (cid:107) B ( H ( R ,q )) (3.83)Thus, H ( R , q ) being a *-subalgebra of B ( H ( R , q )), we can take its closure C ∗ r ( R , q )in the norm topology. By definition this is a separable unital C ∗ -algebra, and wecall it the reduced C ∗ -algebra of H ( R , q ). This is analogous to the reduced C ∗ -algebra of a locally compact group G , which is the completion of C ( G ) for thetopology coming from the regular representation of G .There is also a ”smoother” way to complete an affine Hecke algebra. As atopological vector space, it will consist of rapidly decreasing functions on W withrespect to some length function. For this purpose it is a little unsatisfactory that (cid:96) is 0 on the subgroup Z ( W ) = X + ∩ X − , since this can be the whole of W . Toovercome this inconvenience, let f : X ⊗ R → [0 , ∞ ) be a function such that • f ( X ) ⊂ Z • f ( x + q ) = f ( x ) ∀ x ∈ X ⊗ R , q ∈ Q ⊗ R • f induces a norm on X ⊗ R (cid:14) Q ⊗ R ∼ = Z ( W ) ⊗ R Now we define for w ∈ W N ( w ) := (cid:96) ( w ) + f ( w (0)) (3.84)so that for any w, v ∈ W, u ∈ W aff , ω ∈ Ω N ( uω ) = N ( ωu ) = (cid:96) ( u ) + N ( ω ) (3.85) N ( vw ) ≤ N ( v ) + N ( w ) (3.86)Because Z ( W ) + Q is of finite index in X , the set Ω (cid:48) = { w ∈ W : N ( w ) = 0 } isfinite. Moreover, since W is the semidirect product of a finite group and an abeliangroup, it is of polynomial growth, and different choices of f lead to equivalentlength functions N . Now we can define for any n ∈ N the norm p n (cid:16) (cid:88) w ∈ W x w N w (cid:17) := sup w ∈ W | x w | ( N ( w ) + 1) n (3.87)The completion S ( R , q ) of H ( R , q ) with respect to the family of norms { p n } n ∈ N isa nuclear Fr´echet space. It consists of all possible infinite sums x = (cid:80) w ∈ W x w N w such that p n ( x ) < ∞ ∀ n ∈ N . Opdam [98, Section 6.2] proved that these normsbehave reasonably with respect to multiplication: Theorem 3.12
There exist C q > , d ∈ N such that ∀ x, y ∈ S ( R , q ) , n ∈ N (cid:107) x (cid:107) o ≤ C q p d ( x ) (3.88) p n ( xy ) ≤ C q p n + d ( x ) p n + d ( y ) (3.89) In particular S ( R , q ) is a unital locally convex *-algebra, and it is contained in C ∗ r ( R , q ) . .2. Representation theory 81The proof of this theorem uses heavy machinery, namely the spectral decomposi-tion of the trace τ , which we will state in Theorem 3.24.4. Closer examination ofthat proof shows that we can take d = rk( X ) + | W | + 1. In Section 5.2 we willuse more elementary tools to prove a generalization of this theorem, resulting ina smaller d .We call S ( R , q ) the Schwartz algebra of H ( R , q ). In Section 4.2 we will see thatthis construction is analogous to the Schwartz algebra of a reductive p -adic group.Considering Schwartz completions of parabolic subalgebras, we see that S ( R P , q P )is still a quotient of S ( R P , q P ), but that the latter algebra is in general no longercontained in S ( R , q ). The same holds for the respective reduced C ∗ -algebras.From the work of Casselman [29, § H -representations extend continuously to certain completions.Indeed it is shown in [98, Lemma 2.20] that Lemma 3.13
For an irreducible H -representation ( π, V ) the following conditionsare equivalent, and summarized by saying that π belongs to the discrete series: • ( π, V ) is a subrepresentation of the left regular representation ( λ, H ( R , q )) • all matrix coefficients of ( π, V ) are in H ( R , q ) • | x ( t ) | < for every A -weight t of V and every x ∈ X + \ C ∗ -algebra C ∗ r ( R , q ). Because this is a Hilbert algebra, a suitable ver-sion of [41, Proposition 18.4.2] shows that π is an isolated point in its spectrumPrim (cid:0) C ∗ r ( R , q ) (cid:1) . Moreover, since C ∗ r ( R , q ) is unital, its spectrum is compact [41,Proposition 3.18], so there can be only finitely many inequivalent discrete seriesrepresentations. On the other hand, they can only exist if X + ∩ X − = 0, i.e. ifthe root datum R is semisimple.Let us mention two important examples. If W = W aff and q ( s ) > ∀ s ∈ S aff then π St : N w → ( − (cid:96) ( w ) q ( w ) − / defines a discrete series representation. This is called the Steinberg representationof H ( R , q ). Contrarily, if q ( s ) < ∀ s ∈ S aff then the representation π triv : N w → q ( w ) / is discrete series. This is known as the trivial representation of H ( R , q ), becauseit is a deformation of the trivial representation of W .A linear functional f : H → C is tempered if there exist C, N ∈ (0 , ∞ ) suchthat for all w ∈ W | f ( N w ) | ≤ C (1 + N ( w )) N The collection of all tempered functionals is the linear dual of S ( R , q ). From [98,Lemma 2.20] we get the following characterization: Lemma 3.14
For ( π, V ) ∈ Rep( H ) the following are equivalent: • π extends continuously to S ( R , q ) • every matrix coefficient of ( π, V ) is a tempered functional • V is a tempered H -representation Suppose that our root datum is a product R = R × R . It is clear from (3.87)that the Schwartz completion respects the decomposition (3.42), so S ( R , q ) ∼ = S ( R , q ) (cid:98) ⊗ S ( R , q ) (3.90)An isomorphism like C ∗ r ( R , q ) ∼ = C ∗ r ( R , q ) ⊗ t C ∗ r ( R , q )should also hold, but here the particular topological tensor product ⊗ t is probablydictated by the very isomorphism.Let q be the label function that is identically 1 and T u the unitary or realcompact part of T . For any R the above constructions reduce to the well-knownalgebras H ( R , q ) = C [ W ] = C [ X ] (cid:111) W ∼ = O ( T ) (cid:111) W S ( R , q ) = S ( W ) ∼ = S ( X ) (cid:111) W ∼ = C ∞ ( T u ) (cid:111) W C ∗ r ( R , q ) = C ∗ r ( W ) ∼ = C ∗ r ( X ) (cid:111) W ∼ = C ( T u ) (cid:111) W (3.91)where S ( X ) denotes the space of Schwartz functions on X . The representationtheory of these algebras is not very complicated, and can be obtained from classicalresults which go back to Frobenius and Clifford [31]. Theorem 3.15
1. The W -module I t = Ind WX ( C t ) is completely reducible forany t ∈ T .2. I t is unitary if and only if t ∈ T u , in which case it is also tempered andanti-tempered.3. The number of inequivalent irreducible summands of I t is exactly the numberof conjugacy classes in the isotropy group W ,t .Proof.
1. As an X -representation I t ∼ = (cid:77) w ∈ W C wt and the T w act on I t by permuting these onedimensional subspaces. Hence thereis a natural bijection between subrepresentations of the left regular representationof W ,t and subrepresentations of I t . It is given explicitly by V → Ind W W ,t V , with X acting on the subspace T w V by the character wt . Since C [ W ,t ] is completelyreducible, so is I t .2 will be a special case of Proposition 3.17.3 follows from counting the irreducible representations of the finite group W ,t . (cid:50) .3. The Fourier transform 83 Now we really delve into the representation theory of affine Hecke algebras. This isa very complicated subject and we will barely prove anything. In fact this sectionis more like an overview of some of the work of Delorme and Opdam [39, 40, 98].In principle we mention only those things that we use later in some way or an-other, but this turns out to be a lot. The highlights are undoubtedly the conciseTheorems 3.24 and 3.25, which describe the Fourier transform of affine Hecke al-gebras and their completions. Most of the other things are technicalities that canbe skipped on first reading.For any t ∈ T P there is a surjective algebra homomorphism φ t : H P → H P which is the identity on H ( W , q ) and sends θ x to t ( x ) θ x P , where x P is the imageof x in X P = X/ (cid:0) X ∩ ( P ∨ ) ⊥ (cid:1) . If σ ∈ Rep( H P ) then we can construct the H -representation π ( P, σ, t ) = Ind HH P ( σ ◦ φ t ) (3.92)Because A is in general not a *-subalgebra of H , it is not immediately clear whetherthis procedure preserves unitarity or temperedness of representations. Recall from[97, Propostion 1.12] that Lemma 3.16
Let w be the longest element of W . For x ∈ X we have θ ∗ x = T w θ − w x T − w = N w θ − w x N − w In particular for x ∈ X ∩ (cid:0) P ∨ (cid:1) ⊥ we have θ ∗ x = θ − x in H P , so φ t preservesthe * if and only if t ∈ T Pu . Similarly the inclusion H P → H is in general not*-preserving. Nevertheless Proposition 3.17
Let σ be an irreducible H P -representation and t ∈ T P .1. π ( P, σ, t ) is unitary if and only if σ is unitary and t ∈ T Pu π ( P, σ, t ) is (anti-)tempered if and only if σ is (anti-)tempered and t ∈ T Pu Proof.
The ”if” statements are [98, Propositions 4.19 and 4.20], so we prove the”only if” parts.1. Clearly σ ◦ φ t can only be unitary if σ is. If now t ∈ T P \ T Pu then there is an x ∈ X ∩ (cid:0) P ∨ (cid:1) ⊥ with | t ( x ) | (cid:54) = 1. Hence σ ◦ φ t ( θ ∗ x ) = σ ◦ φ t ( θ − x ) = σ ( t ( − x )) = t ( x ) − (cid:54) = t ( x ) = σ ( t ( x )) ∗ = (cid:0) σ ◦ φ t ( θ x ) (cid:1) ∗
2. Again it is obvious that σ needs to be tempered for σ ◦ φ t to be so. If t and x are as above, then either | t ( x ) | > | t ( − x ) | >
1, so by Lemma 3.14 σ ◦ φ t is not4 Chapter 3. Affine Hecke algebrastempered. This argument also applies in the anti-tempered case. (cid:50) Let ∆ P be the finite set of equivalence classes of discrete series representationsof H P = H ( R P , q P ). We pick a representative in every class and confuse ∆ P withthis set of representations. Write∆ = { ( P, δ ) : P ⊂ F , δ ∈ ∆ P } and denote by Ξ the complex algebraic variety consisting of all triples ( P, δ, t ) with P ⊂ F , δ ∈ ∆ P and t ∈ T P . Similarly let Ξ u be the compact smooth manifold ofall triples ( P, δ, t ) ∈ Ξ with t ∈ T Pu .Let V δ be the representation space of δ , endowed with the inner product (cid:104)· , ·(cid:105) δ ,and define an inner product on H (cid:0) W P (cid:1) ⊗ C V δ by (cid:104) x ⊗ u , y ⊗ v (cid:105) = τ ( x ∗ y ) (cid:104) u , v (cid:105) δ x, y ∈ H (cid:0) W P (cid:1) u, v ∈ V δ (3.93)With ξ = ( P, δ, t ) ∈ Ξ we associate the parabolically induced representation π ( ξ ) = π ( P, δ, t ) = Ind HH P ( δ ◦ φ t ) (3.94)We realize it on H (cid:0) W P (cid:1) ⊗ C V δ with the inner product (3.93).For P = ∅ we have H P ∼ = C . We denote its unique discrete series representationby δ ∅ . Note that π ( ∅ , δ ∅ , t ) is just the principal series representation I t . .We gather all parabolically induced representations in the following vectorbundle over Ξ : V Ξ = (cid:91) ( P,δ ) ∈ ∆ T P × H (cid:0) W P (cid:1) ⊗ V δ (3.95)Sometimes we will write ξ ∈ Ξ as (
P, W P r, δ, t ) to indicate that W P r ∈ T P /W P is the central character of δ . Then the central character of π ( ξ ) is W rt ∈ T /W .If we let t run over T P , we obtain a coset rT P of the subtorus T P in T . Thecollection of all such cosets, for different ( P, δ ) ∈ ∆ and different representativesof their central characters, has some special properties. It can be described com-binatorially with the help of Macdonald’s c -function, whose construction we recallnow. For a long root α ∈ R we put c α = (cid:16) q − / α ∨ θ − α/ (cid:17) (cid:16) − q − / α ∨ q − α ∨ θ − α/ (cid:17) (cid:0) − θ − α (cid:1) − α ∈ R (3.96)This is a rational function on T , i.e. an element of the quotient field Q ( A ) of A .Strictly speaking, for α ∈ R ∩ R this formula is ill-defined, because there is nosuch thing as θ − α/ . However in that case q α ∨ = 1 by our convention (3.31), sowe can interpret the above formula as c α = (cid:0) − q − α ∨ θ − α (cid:1) (cid:0) − θ − α (cid:1) − α ∈ R ∩ R (3.97)Macdonald’s c -function is the product c ( t ) = (cid:89) α ∈ R +1 c α ( t ) (3.98).3. The Fourier transform 85Suppose that L ⊂ T is a coset of a subtorus T L , and let R L = { α ∈ R : α ( T L ) = 1 } (3.99)be the set of long roots that are constant on T L . Then T L = { t ∈ T : x ( t ) = 1 if x ∈ Q R L ∩ X } = Hom Z (cid:0) X/ ( Q R L ∩ X ) , C × (cid:1) (3.100)Hence θ − α/ ( L ) (or θ − α ( L ) if α / ∈ R ) is well-defined. Put R pL = (cid:8) α ∈ R L : (cid:0) q − / α ∨ θ − α/ ( L ) (cid:1)(cid:0) − q − / α ∨ q − α ∨ θ − α/ ( L ) (cid:1) = 0 (cid:9) R zL = { α ∈ R L : 1 − θ − α ( L ) = 0 } (3.101)By construction the pole order of the rational function t → c − ( t ) c − ( t − ) (3.102)along L is | R pL | − | R zL | . We say that L is a residual coset if | R pL | − | R zL | = dim( T ) − dim( L ) (3.103)A residual coset of dimension 0 is also called a residual point. More or less ev-erything there is to know about residual cosets can be found in [58, Section 4]and [98, Appendix A]. The collection of residual cosets is finite and W -invariant.They have been classified completely, and it turns out that a coset of a subtorusis residual if and only if its pole order for (3.102) is maximal, given its dimension.From (3.101) we see that the residual cosets depend algebraically on q . Inparticular the number of residual cosets is maximal for all q in a certain Zariskiopen subset of the parameter space of all possible q ’s. Hence it makes sense to call q generic if and only if the number of residual cosets attains its maximum at q .Define T L = (cid:8) t ∈ T : t ( x ) = 1 if x ∈ ( R ∨ L ) ⊥ ∩ X (cid:9) = Hom Z (cid:0) X (cid:14)(cid:0) ( R ∨ L ) ⊥ ∩ X (cid:1) , C × (cid:1) (3.104)This is a subtorus of T and K L = T L ∩ T L is a finite group. Notice that if L is acoset of T P for some P ⊂ F , then R L = R P , T L = T P and K L = K P . Pick any r L ∈ T L ∩ L and consider r L | r L | − ∈ T L,u ∩ L Since r L is unique up to K L ∩ T Lu , it follows that the tempered form L temp = r L | r L | − T Lu (3.105)is independent of the choice of r L . The tempered forms of residual cosets tend tobe disjoint, cf. [98, Theorem A.18] : Theorem 3.18
Let L and L be residual cosets. If L temp1 ∩ L temp2 (cid:54) = ∅ then L = w ( L ) for some w ∈ W . Theorem 3.19
The collection of residual cosets is precisely (cid:8) rT P : ( P, W P r, δ, t ) ∈ Ξ (cid:9) The tempered form of L = rT P is L temp = rT Pu . This theorem implies that for every W -orbit of residual points there is at leastone discrete series representation with that central character. In many cases thereis in fact exactly one such representation: Proposition 3.20
1. Let t ∈ T be a residual point such that the orbit W t contains exactly | W | points. Then there is, up to equivalence, a uniquediscrete series representation with this central character.2. If the root system R is of type A n then 1. applies, and the representationin question is onedimensional.Proof. For 1 see [121, Corollary 1.2.11]. For 2 we use the classification of the resid-ual points for type A n in [58, Proposition 4.1]. This shows that we can apply 1.To show that the resulting representation has dimension one, we just construct it.Since all simple roots are conjugate, we necessarily have an equal label function.If W = W aff then either π St or π triv is a onedimensional discrete series represen-tation. Which one depends on whether q ( s ) > q ( s ) < s ∈ S aff , seepage 81. If W (cid:54) = W aff then, given the central character, there is unique way toextend this representation to C ∗ r ( R , q ) . (cid:50) However, in general there may be more than one discrete series representationwith a given central character. This is known to happen already for R of type B , for certain label functions, see Section 6.4.Following [98, Definition 3.24] we define a radical and a residual algebra at t ∈ T : H t = H ( R , q ) / Rad t Rad t = ∩ π ker π (3.106)where π runs over the irreducible representations of H ( R , q ) with the followingtwo properties : • the central character of π is W t • π extends to C ∗ r ( R , q )Clearly Rad t contains all elements of Z that vanish at t , so H t is a finite dimen-sional C ∗ -algebra whose irreducible representations correspond to the irreduciblerepresentations of C ∗ r ( R , q ) with central character W t . By Theorem 3.19 the col-lection of discrete series representations of H ( R , q ) is precisely the union, over all.3. The Fourier transform 87residual points t , of the irreducible representations of H t .As one would expect, two parabolically induced representations can be relatedif they have the same central character. Let us try to clarify when and how thisoccurs. For P, Q ⊂ F write W ( P, Q ) = { n ∈ W : n ( P ) = Q } (3.107)For such n there is a *-algebra isomorphism ψ n : H P → H Q , defined by ψ n ( N w θ x ) = N nwn − θ nx w ∈ W P , x ∈ X (3.108)Similarly for k ∈ K P there is a *-algebra automorphism ψ k : H P → H P , definedby ψ k ( N w θ x ) = k ( x ) N w θ x w ∈ W P , x ∈ X (3.109)These maps descend to isomorphisms ψ n : H P → H Q and ψ k : H P → H P .Let W be the finite groupoid, over the power set of F , with W P Q = K Q × W ( P, Q ) (3.110)( k , n )( k , n ) = ( k n ( k ) , n n ) ( k , n ) ∈ W P Q , ( k n ) ∈ W P (cid:48) P (3.111)For kn ∈ W P Q and δ ∈ ∆ P , let Ψ kn ( δ ) be the equivalence class of the discreteseries representation δ ◦ ψ − n ◦ ψ − k of H Q . Then there exists a unitary isomorphism˜ δ kn : V δ → V Ψ kn ( δ ) such that˜ δ kn ◦ δ ( ψ − n ψ − k h ) = Ψ kn ( δ )( h ) ◦ ˜ δ kn h ∈ H Q (3.112)By Schur’s lemma ˜ δ kn is unique up to a complex number of absolute value 1, butwhether or not there is a canonical way to choose these scalars is an open problem.This is a subtle point to which we will return later.The above maps induce an action of W on Ξ : kn ( P, W P r, δ, t ) = (cid:0) Q, W Q n ( r ) , Ψ kn ( δ ) , kn ( t ) (cid:1) (3.113)We would like to lift ˜ δ g to an intertwiner π ( g, ξ ) between π ( ξ ) and π ( gξ ). For k ∈ K p this is easy, we can simply define π ( k, ξ ) = id H ( W P ) ⊗ ˜ δ k : H ⊗ H P V δ → H ⊗ H P V δ (3.114)This is a well-defined intertwiner since, for all h ∈ H , h (cid:48) ∈ H P , v ∈ V : π ( k, ξ )( hh (cid:48) ⊗ v ) = hh (cid:48) ⊗ ˜ δ k ( v )= h ⊗ Ψ k ( δ )( φ kt h (cid:48) ) (cid:0) ˜ δ k v (cid:1) = h ⊗ Ψ k ( δ ) (cid:0) ψ k ( φ t h (cid:48) ) (cid:1)(cid:0) ˜ δ k v (cid:1) = h ⊗ ˜ δ k (cid:0) δ ( φ t h (cid:48) ) v (cid:1) = π ( k, ξ ) (cid:0) h ⊗ δ ( φ t h (cid:48) ) v (cid:1) (3.115)8 Chapter 3. Affine Hecke algebrasOn the other hand, for w ∈ W it is a tricky business, which is taken care of in[98, § U be a W -invariant subset of T , open in the analytic topology, and let C an ( U ) and Z an ( U ) be the algebras of, respectively, holomorphic functions on U and W -invariant holomorphic functions on U . There are natural injections A → C an ( U ) and Z → Z an ( U ), so we can construct H an ( U ) = Z an ( U ) ⊗ Z H = C an ( U ) ⊗ A H (3.116)Similarly we define the algebras C me ( U ) and Z me ( U ) of ( W -invariant) meromor-phic functions on U , and H me ( U ) = Z me ( U ) ⊗ Z H = C me ( U ) ⊗ A H (3.117)It follows from Theorem 3.3.1 that H an ( U ) and H me ( U ) are free modules overrespectively C an ( U ) and C me ( U ), with bases { N w : w ∈ W } . We define a * onthese algebras by f ∗ = T w f − w T − w = N w f − w N − w f ∈ C me ( U ) (3.118)By Lemma 3.16 this extends the * on H ( R , q ).A finite dimensional H -representation extends to H an ( U ) if and only if allits Z -weights are in U/W . Applying this to H P , we see that δ ◦ φ t extends to H P,an ( U ) if and only if W P rt ∈ U/W P . The proof of these claims can be foundon [98, p. 582].Pick t ∈ T and consider a set B ⊂ Lie( T ) = t ⊗ R C which satisfies the followingconditions: Condition 3.21 B is an open ball centred around ∈ t ⊗ R C ∀ α ∈ R nr , b ∈ B : |(cid:61) ( α ( b )) | < π (so exp : B → exp( B ) is a diffeomorphism)3. Write U t = t exp( B ) . If w ∈ W and wU t ∩ U t (cid:54) = ∅ , then wt = t ∀ u ∈ U t we have R p { u } ∪ R z { u } ⊂ R p { t } ∪ R z { t } Recall that R p { t } ∪ R z { t } is the set of roots α ∈ R for which t is a critical point ofthe rational function c α . Notice that these conditions are almost the same as [98,Conditions 4.9]. In fact the only difference is that we replaced some statementswith B by slightly stronger versions with B . This is not an essential difference, itonly makes it easier in Section 5.3 to see that certain functions are bounded.By [98, Proposition 4.7] Condition 3.21 can always be fulfilled. This yields analternative description of the representation space of π ( P, W P r, δ, t ) : H (cid:0) W P (cid:1) ⊗ C V δ = H ⊗ H P V δ = H an ( W U rt ) ⊗ H P,an ( W P U rt ) V δ (3.119).3. The Fourier transform 89Let s ∈ S be the simple reflection corresponding to α ∈ F . Consider the followingelement of Q ( A ) ⊗ A H = Q ( Z ) ⊗ Z H : ı s = (cid:0) q / α ∨ + θ − α/ (cid:1) − (cid:0) q / α ∨ q α ∨ − θ − α/ (cid:1) − (cid:0) (1 − θ − α ) T s + ( q α ∨ q α ∨ −
1) + q / α ∨ ( q α ∨ − θ − α/ (cid:1) = (cid:0) T s (1 − θ α ) + ( q α ∨ q α ∨ − θ α + q / α ∨ ( q α ∨ − θ α/ (cid:1)(cid:0) q / α ∨ + θ α/ (cid:1) − (cid:0) q / α ∨ q α ∨ − θ α/ (cid:1) − (3.120)For α ∈ F ∩ R this simplifies to ı s = ( q ( s ) − θ − α ) − ((1 − θ − α ) T s + q ( s ) −
1) = ( T s (1 − θ α )+( q ( s ) − θ α )( q ( s ) − θ α ) − Important properties of such elements come from [86, Proposition 5.5] and [98,Lemma 4.1] :
Theorem 3.22
The map S → Q ( A ) ⊗ A H : s → ı s extends to a group homomorphism W → (cid:0) Q ( A ) ⊗ A H (cid:1) × For all w ∈ W , f ∈ Q ( A ) we have ı w f ı w − = f w = f ◦ w − ∈ Q ( A ) (3.121) As Q ( A ) -modules, we can identify Q ( A ) ⊗ A H = (cid:77) w ∈ W ı w Q ( A ) = (cid:77) w ∈ W Q ( A ) ı w If t ∈ T and c − α ( t ) = 0 then by definition ı os α ( t ) = 1. If we combine this with(3.121) we see that it is not specific for simple reflections: β ∈ R , c − β ( t ) = 0 ⇒ ı os β ( t ) = 1 (3.122)With a straightforward but tedious computation one may show that (cid:0) ı w (cid:1) ∗ = T w (cid:89) α ∈ R +1 ∩ w (cid:48) R − (cid:18) c α c − α (cid:19) ı w (cid:48) T − w (3.123)where w (cid:48) = w w − w . It follows more readily from the Bernstein-Lusztig-Zelevinskirelations (3.41) that, for n ∈ W ( P, Q ) and w ∈ W P , ı n − N w ı n = N nwn − (3.124)0 Chapter 3. Affine Hecke algebrasHence we have the following equality in Q ( A ) ⊗ A H : ı n − hı n = ψ n ( h ) h ∈ H P (3.125)For ξ = ( P, W P r, δ, t ) ∈ Ξ we define π ( n, ξ ) : H an ( W U rt ) ⊗ H P,an ( W P U rt ) V δ → H an ( W U rt ) ⊗ H Q,an ( W Q U n ( rt ) ) V Ψ n ( δ ) π ( n, ξ )( h ⊗ v ) = h ı n − ⊗ ˜ δ n ( v ) h ∈ H , v ∈ V δ (3.126)With (3.119) in mind, this map intertwines the H -representations π ( ξ ) and π ( nξ )since, for h ∈ H , h (cid:48) ∈ H P , v ∈ V δ : π ( n, ξ )( hh (cid:48) ⊗ v ) = hh (cid:48) ı n − ⊗ ˜ δ n ( v )= hı n − ( ı n h (cid:48) ı n − ) ⊗ ˜ δ n ( v )= h ı n − ⊗ Ψ n ( δ ) (cid:0) φ n ( t ) ( ı n h (cid:48) ı n − ) (cid:1)(cid:0) ˜ δ n v (cid:1) = h ı n − ⊗ Ψ n ( δ ) (cid:0) φ n ( t ) ( ψ n h (cid:48) ) (cid:1)(cid:0) ˜ δ n v (cid:1) = h ı n − ⊗ Ψ n ( δ ) (cid:0) ψ n ( φ t h (cid:48) ) (cid:1)(cid:0) ˜ δ n v (cid:1) = h ı n − ⊗ ˜ δ n (cid:0) δ ( φ t h (cid:48) ) v (cid:1) = π ( n, ξ ) (cid:0) h ⊗ δ ( φ t h (cid:48) ) v (cid:1) (3.127)For g = kn ∈ W we define the intertwiner π ( g, ξ ) = π ( k, nξ ) π ( n, ξ ) (3.128)Because every ˜ δ g is unique up to a scalar, there is a c ( g , g , δ ) ∈ T such that π ( g , g ξ ) π ( g , ξ ) = c ( g , g , δ ) π ( g g , ξ ) (3.129)Things would simplify considerably if we could arrange that all c ( g , g , δ ) wouldbe 1. In many cases this is indeed possible, but there are some examples for whichit cannot be done.A priori π ( kn, ξ ) is well-defined only on the Zariski-open subset of ( P, δ, T P )where ı n and ı n − are regular. In a worst case scenario this set could even beempty, but fortunately it is assured by [98, Theorem 4.33 and Corollary 4.34] that Theorem 3.23
For any g ∈ W P Q intertwining map π ( g, ξ ) : H (cid:0) W P (cid:1) ⊗ V δ → H (cid:0) W Q (cid:1) ⊗ V Ψ g ( δ ) is rational as a function of t ∈ T P . It is regular on an analytically open neighbor-hood of T Pu . Moreover, if t ∈ T Pu then π ( g, ξ ) is unitary with respect to the innerproducts defined by (3.93) . .3. The Fourier transform 91With this in mind, let Γ rr (Ξ; End( V Ξ )) be the algebra of rational sections ofEnd( V Ξ ) that are regular on Ξ u . It can be described more explicitly asΓ rr (cid:0) Ξ; End( V Ξ ) (cid:1) = (cid:77) ( P,δ ) ∈ ∆ Γ rr (cid:0) T P ; End (cid:0) H (cid:0) W P (cid:1) ⊗ V δ (cid:1)(cid:1) = (cid:77) ( P,δ ) ∈ ∆ (cid:8) f ∈ Q (cid:0) O (cid:0) T P (cid:1)(cid:1) ⊗ End (cid:0) H (cid:0) W P (cid:1) ⊗ V δ (cid:1) : f is regular on T Pu (cid:9) (3.130)Obviously this algebra contains the algebra of polynomial sections O (cid:0) Ξ; End( V Ξ ) (cid:1) = (cid:77) ( P,δ ) ∈ ∆ O (cid:0) T P (cid:1) ⊗ End (cid:0) H (cid:0) W P (cid:1) ⊗ V δ (cid:1) (3.131)Using the standard analytic structure on Ξ, we define the algebras C (Ξ; End( V Ξ ))and C ∞ (Ξ; End( V Ξ )) of continuous (respectively smooth) sections of End( V Ξ ) inthe same way. Furthermore, if µ is a sufficiently ”nice” measure on Ξ then the fol-lowing formula defines a (degenerate) Hermitian inner product on C (Ξ; End( V Ξ )): (cid:104) f , f (cid:105) µ = (cid:90) Ξ tr (cid:0) f ( ξ ) ∗ f ( ξ ) (cid:1) dµ ( ξ ) (3.132)We denote the corresponding Hilbert space completion by L (Ξ , µ ; End( V Ξ )).From the action of the groupoid W on Ξ and the above intertwiners we get anaction of W on Γ rr (Ξ; End( V Ξ )) by algebra homomorphisms :( g · f )( ξ ) = π ( g, g − ξ ) f ( g − ξ ) π ( g, g − ξ ) − (3.133)whenever g − ξ is defined. The average of f over W is p W ( f )( P, δ, t ) = (cid:12)(cid:12) { ( Q, g ) : Q ⊂ F , g ∈ W QP } (cid:12)(cid:12) − (cid:88) Q ⊂ F (cid:88) g ∈W QP ( g · f )( P, δ, t )(3.134)Notice that for f ∈ O (Ξ; End( V Ξ )) and g ∈ W , g · f and p W ( f ) need not lie in C (Ξ; End( V Ξ )). However, if M is a W -stable subset of Ξ on which all the inter-twiners are regular, then there is a well-defined action of W on C ( M ; End( V Ξ )).The same holds for smooth sections if M is a smooth submanifold of Ξ on top ofthis.The Fourier transform is the algebra homomorphism F : H → O (cid:0)
Ξ; End( V Ξ ) (cid:1) F ( h )( ξ ) = π ( ξ )( h ) (3.135)We can extend it continously to various completions of H ( R , q ). After doing so,its image can be described completely with our intertwiners. For the Hilbert spacecompletion H ( R , q ) the following was proved in [98, Theorem 4.43 and Corollary4.45] :2 Chapter 3. Affine Hecke algebras Theorem 3.24
There exists a unique ”Plancherel” measure µ P l on Ξ with theproperties1. the support of µ P l is Ξ u µ P l is W -invariant3. on every component ( P, δ, T P ) µ P l is absolutely continuous with respect tothe Haar measure of T Pu
4. the Fourier transform extends to a bijective isometry F : H ( R , q ) → L (Ξ u , µ P l ; End( V Ξ )) W i.e. µ P l is the Plancherel measure for τ
5. the adjoint map J : L (Ξ u , µ P l ; End( V Ξ )) → H ( R , q ) satisfies J F = id H ( R ,q ) and FJ = p W . The corresponding statements for the Schwartz and C ∗ -completions are [39,Theorem 4.3 and Corollary 4.7] : Theorem 3.25
The Fourier transform induces algebra homomorphisms H ( R , q ) → O (cid:0) Ξ; End( V Ξ ) (cid:1) W S ( R , q ) → C ∞ (cid:0) Ξ u ; End( V Ξ ) (cid:1) W C ∗ r ( R , q ) → C (cid:0) Ξ u ; End( V Ξ ) (cid:1) W The first one is injective, the second is an isomorphism of Fr´echet *-algebras andthe third is an isomorphism of C ∗ -algebras. Some remarkable consequences of this theorem are
Corollary 3.26
1. The centers of S ( R , q ) and C ∗ r ( R , q ) are Z (cid:0) S ( R , q ) (cid:1) ∼ = C ∞ (Ξ u ) W Z (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = C (Ξ u ) W
2. Every irreducible tempered H ( R , q ) -representation ( π, V ) is a direct sum-mand of π ( ξ ) , for some ξ ∈ Ξ u . In particular we can endow V with an innerproduct such that π is unitary and extends to C ∗ r ( R , q ) .3. For any ξ ∈ Ξ and g ∈ W such that gξ is defined, the H ( R , q ) -representations π ( ξ ) and π ( gξ ) have the same irreducible subquotients, counted with multi-plicity. .3. The Fourier transform 93 Proof. S ( R , q )-representation ( π, V ) has a single Z ( S ( R , q ))-weight W ξ ∈ Ξ u / W . Then it is also an irreducible representation of the finite dimensional C ∗ -algebra S ( R , q ) / { h ∈ S ( R , q ) : π ( gξ )( h ) = 0 ∀ g ∈ W} By Theorem 3.25 every irreducible representation of this algebra is a direct sum-mand of a parabolically induced representation π ( gξ ). Again by Theorem 3.25this means that ( π, V ) extends to a (unitary) representation of C ∗ r ( R , q ) .
3. See [40, Proposition 6.1]. We have to show that the characters of π ( ξ ) and π ( gξ ) are equal, i.e. that the function H × T P → C : ( h, t ) → tr π ( P, δ, t )( h ) − tr π ( Q, Ψ g ( δ ) , g ( t ))( h ) (3.136)is identically 0. Because this is a polynomial function of t , it suffices to show thatit is 0 for all t ∈ T Pu . This follows directly from Theorem 3.25. (cid:50) Corollary 3.27
1. An element h ∈ S ( R , q ) is invertible in S ( R , q ) if and onlyif it is invertible in C ∗ r ( R , q ) , which happens if and only if it is invertible in B ( H ( R , q )) .2. The set of invertible elements S ( R , q ) × is open in S ( R , q ) , and inverting isa continuous map from this set to itself.Proof. By [125, Proposition 4.8] and Theorem 3.25 the inclusions S ( R , q ) → C ∗ r ( R , q ) → B ( H ( R , q )) (3.137)are isospectral, which is another way to state 1. Because the set of invertibles isalways open in a Banach algebra, so is S ( R , q ) × ∼ = C ∞ (cid:0) Ξ u ; End( V Ξ ) (cid:1) W ∩ C (cid:0) Ξ u ; End( V Ξ ) (cid:1) × (3.138)Finally, by Theorem 2.9.2 inverting is continuous on S ( R , q ) × . (cid:50) Now we will combine the Langlands classification with the Fourier transformto obtain a finer classification of irreducible H -modules. Although the proofs ofthe following results are mostly in [40], we give them anyway, because we want togeneralize them to Hecke algebras of reductive p -adic groups.For Q ⊂ F , let W Q , Ξ Q , π Q ( ξ ) etcetera denote the same as the correspondingobjects without the superscript Q , but now for the affine Hecke algebra H Q . For ξ = ( P, W P r, δ, t ) ∈ Ξ we put P ( ξ ) = { α ∈ F : | α ( t ) | = 1 } ⊃ P (3.139)We will study π ( ξ ) by induction in stages, first we construct π P ( ξ ) ( ξ ) and thenwe induce that representation to H . The following result from [40, Proposition6.17.1] provides the link with the Langlands classification.4 Chapter 3. Affine Hecke algebras Lemma 3.28
The H P ( ξ ) -representation π P ( ξ ) ( ξ ) is essentially tempered and com-pletely reducible.Proof. By Proposition 3.17 π P ( ξ ) ( P, δ, t | t | − ) is tempered and unitary. From theproof of Lemma 3.5 we see that π P ( ξ ) ( ξ ) ◦ ψ − | t | = π P ( ξ ) ( P, δ, t | t | − ) (3.140)so π P ( ξ ) ( ξ ) is completely reducible. If r , . . . , r d are the A -weights of (3.140),then | t | r , . . . , | t | r d are the A -weights of π P ( ξ ) ( ξ ). But for x ∈ Z P ( ξ ) we have | r i t ( x ) | = | r i ( x ) | , so π P ( ξ ) ( ξ ) is essentially tempered. (cid:50) Similar to the definition of Langlands data, we need a kind of positivity con-dition on Ξ. Thus we say that ξ ∈ Ξ + if | t | ∈ T P, + rs = exp t P, + . Proposition 3.29
Take ξ = ( P, W P r, δ, t ) ∈ Ξ + .1. Let σ be an irreducible direct summand of π P ( ξ ) ( ξ ) . Then ( P ( ξ ) , σ ) ∈ Λ + .2. The functor Ind HH P ( ξ ) induces an isomorphism End H ( π ( ξ )) ∼ = End H P ( ξ ) (cid:0) π P ( ξ ) ( ξ ) (cid:1)
3. The irreducible quotients of π ( ξ ) are precisely the modules L ( P ( ξ ) , σ ) with σ as above.4. Every L ( P ( ξ ) , σ ) has an A -weight t σ such that there exists a root subsystem R σ ⊂ R , of rank | P | , with the properties ∀ α ∈ R +0 ∩ R σ : | t σ ( α ) | < ∀ α ∈ R +0 ∩ (cid:0) R ∨ σ (cid:1) ⊥ : | t σ ( α ) | ≥
5. Suppose that ( ρ, V ) is an irreducible constituent of π ( ξ ) which is not a quo-tient, and that t ρ is an A -weight of V . Then every root subsystem R ρ withthe properties (3.141) has rank > | P | .Proof.
1. By definition r σ = | t | ∈ T P, + rs , so ( P, σ ) ∈ Λ + .2 and 3 follow from 1 and Theorem 3.7.4. By Lemma 3.6 every A -weight of σ is of the form t σ = w ( rt ) with r an A P -weight of δ and w ∈ W P ( ξ ) ∩ W P = (cid:8) w ∈ W P ( ξ ) : w ( P ) ⊂ R +0 (cid:9) By [45, (2.7)] L ( P ( ξ ) , σ ) also has an A -weight of this form. Hence we may take R σ = w ( R P )..3. The Fourier transform 955. By [45, (2.7)] every A -weight of ( ρ, V ) is of the form t ρ = nw ( rt ), with n ∈ W Q \ { e } and w, r, t as in the proof of 4. Clearly, for α ∈ nw ( R + P ) ⊂ n (cid:0) R + Q (cid:1) ⊂ R +0 we have (cid:12)(cid:12) t ρ ( α ) (cid:12)(cid:12) = (cid:12)(cid:12) rt (cid:0) w − n − α (cid:1)(cid:12)(cid:12) < n (cid:54) = e , there exists a β ∈ R +0 with n − ( β ) ∈ R − and β ⊥ n ( P ( ξ )) ∨ . Now P (cid:48) = nw ( P ) ∪ { β } is a linearly independent set of positive rootssuch that | t ρ ( α ) | < ∀ α ∈ P (cid:48) . Therefore any suitable root system R ρ must haverank at least | P (cid:48) | = | P | + 1 . (cid:50) Although it is written down differently, the proof of Proposition 3.29 is rathersimilar to that of the Langlands classification. In particular we use the same ideasas Langlands’ geometric lemmas [80, p. 61-63].
Lemma 3.30
Every ξ ∈ Ξ is W -associate to an element of Ξ + . If ξ , ξ ∈ Ξ + are W -associate, then P ( ξ ) = P ( ξ ) := Q , and π Q ( ξ ) and π Q ( ξ ) are equivalentas H Q -representations.Proof. By [61, Section 1.15] every W -orbit in t contains a unique point in apositive chamber t Q, + . Hence | t | = | t | and P ( ξ ) = P ( ξ ) = Q . From Lemmas3.5 and 3.28 we see that there is a single automorphism ψ | t | = ψ | t | := ψ of H Q such that, for i = 1 , π Q ( ξ i ) ◦ ψ − ∼ = π Q ( ξ (cid:48) i ) where ξ (cid:48) i = ( P i , δ i , t i | t i | − ) ∈ Ξ Qu (3.142)If gξ = ξ for some g ∈ W , then also gξ (cid:48) = ξ (cid:48) . Applying Theorem 3.25 to H Q ,we see that π Q ( ξ (cid:48) ) and π Q ( ξ (cid:48) ) are unitarily equivalent. It follows from this and(3.142) that π Q ( ξ ) and π Q ( ξ ) are equivalent. (cid:50) Theorem 3.31
For every irreducible H -representation π there exists a uniqueassociation class W ( P, δ, t ) ∈ Ξ / W such that the following (equivalent) statementshold :1. π is equivalent to an irreducible quotient of π ( ξ + ) , for some ξ + ∈ Ξ + ∩W ( P, δ, t ) π is equivalent to an irreducible subquotient of π ( P, δ, t ) , and | P | is maximalfor this propertyProof. For 1 we copy [40, Corollary 6.19]. By Theorem 3.7 there is a uniqueLanglands datum (
Q, σ ) ∈ Λ + such that π ∼ = L ( P, σ ), and by Lemma 3.5 σ ◦ ψ − r σ is tempered. Now Theorem 3.25 tells us that there exists a unique associationclass W Q ξ = W Q ( P , δ , t ) ∈ Ξ Qu / W Q such that π ( ξ ) contains σ ◦ ψ − r σ as anirreducible direct summand. Put ξ = ( P , δ , t r σ ) ∈ Ξ Q and, using Lemma 3.306 Chapter 3. Affine Hecke algebrasfor H Q , pick ξ + = ( P , δ , t ) ∈ W Q ξ ∩ Ξ + . Then σ is a direct summand of π Q ( ξ + ),and we see from Proposition 3.29.3 that π is an irreducible subquotient of π ( ξ + ).By Lemma 3.30 and Theorem 3.7 the class W ξ ∈ Ξ / W is unique for this property.Suppose that π is also an irreducible subquotient of π ( ξ ) = π ( P , δ , t ), where | P | ≥ | P | . By Corollary 3.26.2 we may assume that ξ ∈ Ξ + . Comparing parts 4and 5 of Proposition 3.29 we see that in fact π must be equivalent to an irreduciblequotient of π ( ξ ). But then ξ is W -associate to ξ + by the above. Because theclass W ξ is unique for both properties 1 and 2, this also shows that 1 and 2 areequivalent. (cid:50) . We prove comparison theorems between the periodic cyclic homology of an affineHecke algebra, that of its Schwartz completion, and the K -theory of its C ∗ -completion. We reap some fruits from our previous labor in the sense that thetechnicalities are limited, although still substantial.Let H ( R , q ) be an affine Hecke algebra. Observe that by Theorems 2.27 and3.25 the Chern character gives an isomorphism K ∗ ( S ( R , q )) ⊗ C ∼ −−→ HP ∗ ( S ( R , q )) (3.143)However, we cannot apply Theorem 2.29 to H ( R , q ), since the action of thegroupoid W on O (Ξ; End( V Ξ )) is by rational intertwiners, which may have polesoutside Ξ u . Therefore the proof of the next theorem will involve several steps. Theorem 3.32
The inclusion H ( R , q ) → S ( R , q ) induces an isomorphism HP ∗ ( H ( R , q )) ∼ −−→ HP ∗ ( S ( R , q )) Proof.
We start by constructing stratifications of the primitive ideal spectra of H ( R , q ) and S ( R , q ). Choose an increasing chain ∅ = ∆ ⊂ ∆ ⊂ · · · ⊂ ∆ n = ∆of W -invariant subsets of ∆ with the properties • if ( P, δ ) ∈ ∆ i and | Q | > | P | then ∆ Q ⊂ ∆ i • the elements of ∆ i \ ∆ i − form exactly one association class for the actionof W To this correspond two decreasing chains of ideals H = I ⊃ I ⊃ · · · ⊃ I n = 0 S = J ⊃ J ⊃ · · · ⊃ J n = 0 I i = { h ∈ H : π ( P, t, δ )( h ) = 0 if ( P, δ ) ∈ ∆ i , t ∈ T P } J i = { h ∈ S : π ( P, t, δ )( h ) = 0 if ( P, δ ) ∈ ∆ i , t ∈ T Pu } (3.144).4. Periodic cyclic homology 97For every i pick an element ( P i , δ i ) ∈ ∆ i \ ∆ i − , let W i be the stabilizer of ( P i , δ i )in W and write V i = V π ( P i ,t,δ i ) . By Theorem 3.25 the Fourier transform givesisomorphisms J i − /J i ∼ = C ∞ ( T P i u ; End V i ) W i (3.145)On the other hand, by Theorem 3.31 the primitive ideal spectrum of ( I i − /I i )corresponds to the inverse image of ∆ i \ ∆ j under the projection Ξ → ∆. Moreoverthe induced map Prim( I i − /I i ) → W i \ T P i is continuous. (In fact it is the centralcharacter map for this algebra.) By Lemma 2.3 it suffices to show that eachinclusion I i − /I i → J i − /J i (3.146)induces an isomorphism on periodic cyclic homology. Therefore we zoom in on J i − /J i . By Theorem 3.23 we can extend the action of W i on C ∞ ( T P i u ; End V i )to a neighborhood T (cid:48) of T P i u . We may take T (cid:48) W i -equivariantly diffeomorphic to T P i u × [ − , dim T Piu . Because [ − ,
1] is compact and contractible, we can make theinner product on V i depend on t ∈ T (cid:48) in a smooth way, such that the intertwiners π ( g, P i , t, δ i ) are unitary on all of T (cid:48) . To avoid some technical difficulties we wantto replace J i − /J i by C ∞ ( T (cid:48) ; End V i ) W , but this needs some justification. Lemma 3.33
The inclusion T P i u → T (cid:48) and the Chern character induce isomor-phisms HP ∗ ( J i − /J i ) ∼ ←−− HP ∗ (cid:0) C ∞ ( T (cid:48) ; End V i ) W i (cid:1) ∼ −−→ K ∗ (cid:0) C ( T (cid:48) ; End V i ) W i (cid:1) ⊗ C Proof.
The second isomorphism follows directly from Theorems 2.27 and 2.13. Forthe first one, we pick a W i -equivariant triangulation Σ → T P i u and we construct U σ and U σ as on page 55. Using the projection p u : T (cid:48) → T P i u we get a closedcover of T (cid:48) : { T (cid:48) σ : σ simplex of Σ } T (cid:48) σ = p − u ( U σ ) ∼ = U σ × [ − , dim T Piu
From the proof of Theorem 2.27 we see that it suffices to show that for any simplex σ we have HP ∗ (cid:0) C ∞ ( U σ , D σ ; End V i ) W σ (cid:1) ∼ = HP ∗ (cid:0) C ∞ ( T (cid:48) σ , p − u ( D σ ); End V i ) W σ (cid:1) (3.147)where W σ is the stabilizer of σ in W i . Well, U σ \ D σ is W σ -equivariantlycontractible by construction, and it is an equivariant deformation retract of T (cid:48) σ \ p − u ( D σ ) = p − u ( U σ \ D σ ). So we are in the setting of Lemma 2.26 and wemay use its proof. It says that there exist a finite central extension G of W σ anda linear representation G → GL ( V i ) : g → u g such that the Fr´echet algebras in (3.147) are isomorphic to C ∞ ( U σ , D σ ; End V i ) G (3.148) C ∞ ( T (cid:48) σ , p − u ( D σ ); End V i ) G (3.149)8 Chapter 3. Affine Hecke algebrasThe G -action on these algebras is given by g ( f )( t ) = u g f ( g − t ) u − g where we simply lifted the action of W σ on T (cid:48) σ to G .It is clear that the retraction T (cid:48) σ → U σ induces a diffeotopy equivalence be-tween (3.148) and (3.149), so it also induces the required isomorphism (3.147). (cid:50) Consider the finite collection L of all irreducible components of ( T P i ) w , as w runs over W i . These are all cosets of complex subtori of T P i and they havenonempty intersections with T P i u . Extend this to a collection { L j } j of cosetsof subtori of T P i by including all irreducible components of intersections of anynumber of elements of L . Because the action α i of W i on T P i is algebraicdim (cid:0) (cid:0) T P i (cid:1) g ∩ (cid:0) T P i (cid:1) w (cid:1) < max (cid:8) dim (cid:0) T P i (cid:1) g , dim (cid:0) T P i (cid:1) w (cid:9) if α i ( w ) (cid:54) = α i ( g ). Define W i -stable submanifolds T m = (cid:91) j : dim L j ≤ m L j T (cid:48) m = T m ∩ T (cid:48) and construct the following ideals A m = { h ∈ I i − /I i : π ( P i , t, δ i )( h ) = 0 if t ∈ T m } B m = C ∞ ( T (cid:48) , T (cid:48) m ; End V i ) W i C m = C ( T (cid:48) , T (cid:48) m ; End V i ) W i (3.150)Now we have A n = B n = C n = 0 for n ≥ dim T P i and A n = I i − /I i B n = C ∞ ( U ; End V i ) W i C n = C ( U ; End V i ) W i for n < A m − /A m → B m − /B m induce isomorphisms on HP ∗ , so let us compute the periodic cyclic homologies ofthese quotient algebras.Because T m is an algebraic subvariety of T P i the spectrum of A m − /A m consistsprecisely of the irreducible representations of I i − /I i with tempered central char-acter in ( P i , T m \ T m − , δ i ). We let r i ( t ) be the number of π ∈ Prim( I i − /I i ) corre-sponding to ( P i , t, δ i ). From Theorem 3.7 and we see that r i ( t | t | s ) = r i ( t ) ∀ s > − r i ( t | t | − ) = r i ( t ) if the stabilizers in W i of t and t | t | − are equal. Choose a minimal subset { L m,k } k of L such that every m -dimensionalelement of L is conjugate under W i to a L m,k . Let W m,k be the stabilizer of L m,k .4. Periodic cyclic homology 99in W i and write r k = r i ( t ) for some t ∈ L m,k \ T P i u . By construction W m,k actsfreely on L m,k \ T m − , and the spectrum of A m − /A m is homeomorphic to X m \ Y m := (cid:71) k r k (cid:71) l =1 ( L m,k \ T m − ) / W m,k = (cid:71) k r k (cid:71) l =1 ( L m,k / W m,k ) \ (cid:0) ( L m,k ∩ T m − ) / W m,k (cid:1) These are separable algebraic varieties, so the morphisms of finite type algebras A m − /A m ← Z ( A m − /A m ) → O ( X m , Y m ) (3.151)are spectrum preserving. Thus from Theorems 2.6 and 2.7 we get natural isomor-phisms HP ∗ ( A m − /A m ) ∼ = HP ∗ (cid:0) O ( X m , Y m ) (cid:1) → ˇ H ∗ ( X m , Y m ; C ) (3.152)On the other hand, by [128, Th´eor`eme IX.4.3] the extension0 → C ∞ ( T (cid:48) , T (cid:48) m ; End V i ) → C ∞ ( T (cid:48) ; End V i ) → C ∞ ( T (cid:48) m ; End V i ) → W i is finite the same holds for0 → B m → C ∞ ( T (cid:48) ; End V i ) W i → C ∞ ( T (cid:48) m ; End V i ) W i → HP ∗ ( B m ) ∼ ←−− K ∗ ( B m ) ⊗ C ∼ −−→ K ∗ ( C m ) ⊗ C (3.153)The spectrum of C m − /C m is X (cid:48) m \ Y (cid:48) m := ( X m ∩ T (cid:48) / W i ) \ ( Y m ∩ T (cid:48) / W i )= (cid:71) k r k (cid:71) l =1 ( L m,k ∩ T (cid:48) m ) / W m,k \ ( L m,k ∩ T (cid:48) m − ) / W m,k These are locally compact Hausdorff spaces, so the C ∗ -algebra homomorphisms C m − /C m ← Z ( C m − /C m ) ∼ −−→ C ( X (cid:48) m , Y (cid:48) m ) (3.154)are spectrum preserving. By construction the stabilizer in W i of t ∈ T (cid:48) is constanton the connected components of T (cid:48) m \ T (cid:48) m − , so by the continuity of the intertwiners π ( g, P i , t, δ i ) the vector space C m − /C m is a projective module over C ( X (cid:48) m , Y (cid:48) m ).Thus by Proposition 2.21 (for K ∗ ( · ) ⊗ Q ) (3.154) induces isomorphisms on K -theory with rational coefficients. From this and Theorems 2.27 and 2.13 we obtain00 Chapter 3. Affine Hecke algebrasnatural isomorphisms HP ∗ ( B m − /B m ) ∼ = K ∗ ( C m − /C m ) ⊗ C ∼ = K ∗ ( Z ( C m − /C m )) ⊗ C ∼ = K ∗ ( C ( X (cid:48) m , Y (cid:48) m )) ⊗ C ∼ = K ∗ ( C ∞ ( X (cid:48) m , Y (cid:48) m )) ⊗ C ∼ = HP ∗ ( C ∞ ( X (cid:48) m , Y (cid:48) m )) ∼ = ˇ H ∗ ( X (cid:48) m , Y (cid:48) m ; C ) (3.155)From (3.152) - (3.155) we construct the commutative diagram HP ∗ ( A m − /A m ) ∼ = HP ∗ (cid:0) O ( X m , Y m ) (cid:1) → ˇ H ∗ ( X m , Y m ; C ) ↓ ↓ ↓ HP ∗ ( B m − /B m ) ∼ = HP ∗ (cid:0) C ∞ ( X (cid:48) m , Y (cid:48) m ) (cid:1) → ˇ H ∗ ( X (cid:48) m , Y (cid:48) m ; C ) (3.156)The pair ( X (cid:48) m , Y (cid:48) m ) is a deformation retract of ( X m , Y m ), so all maps in this dia-gram are isomorphisms. Working our way back up, using excision, we find thatalso HP ∗ ( I i − /I i ) → HP ∗ (cid:0) C ∞ ( T (cid:48) ; End V i ) W i (cid:1) → HP ∗ ( J i − /J i )and finally HP ∗ ( H ( R , q )) → HP ∗ ( S ( R , q ))are isomorphisms. (cid:50) Note that Theorem 3.32 is in accordance with our earlier results for directproducts of root data. If R = R × R then by (3.42) and Theorem 2.5 we have HP ∗ ( H ( R , q )) ∼ = HP ∗ ( H ( R , q )) ⊗ HP ∗ ( H ( R , q )) (3.157)while by (3.90), Theorem 3.25 and Corollary 2.28 HP ∗ ( S ( R , q )) ∼ = HP ∗ ( S ( R , q )) ⊗ HP ∗ ( S ( R , q )) (3.158)We can pursue the path of (3.143) and Theorem 3.32 a little further. Let k beany (unital) subring of C containing { q ( w ) : w ∈ W } . As on page 63 we considerthe extended Iwahori-Hecke algebra H k ( R , q ). It makes sense to take its periodiccyclic homology in the category of k -algebras. This is a k -module which we denoteby HP ∗ ( H k ( R , q ) | k ). Theorem 3.34
There are natural isomorphisms HP ∗ ( H k ( R , q ) | k ) ⊗ k C ∼ −−→ HP ∗ ( H ( R , q )) ∼ −−→ HP ∗ ( S ( R , q )) ∼ ←−− K ∗ ( S ( R , q )) ⊗ Z C ∼ −−→ K ∗ ( C ∗ r ( R , q )) ⊗ Z C .4. Periodic cyclic homology 101 Proof.
By [26, Proposition IX.5.1] we have HH n ( H k ( R , q ) | k ) ⊗ k C ∼ = HH n ( H ( R , q )) (3.159)where HH n ( · | k ) means Hochschild homology in the category of k -algebras. Nowthe first isomorphism follows from [81, Proposition 5.1.6]. The second isomorphismis Theorem 3.32 and the third was already noticed in (3.143). Finally, the fourthisomorphism is a consequence of Theorem 2.13. (cid:50) Apparently this is an important invariant of the labelled root datum ( R , q ). Bythe way, we really need complex coefficients. It does not follow from Theorem 3.34that HP ∗ ( H k ( R , q ) | k ) and K ∗ ( C ∗ r ( R , q )) ⊗ Z k are naturally isomorphic, we merelyknow that they have the same (finite) rank as k -modules. In general there is noreason why the image of a class in K ∗ ( C ∗ r ( R , q )) should land in HP ∗ ( H k ( R , q ) | k )under the composition of the above isomorphisms.02 Chapter 3. Affine Hecke algebras hapter 4 Reductive p -adic groups Iwahori and Matsumoto were the first to recognize that any finite group with a BN -pair gives rise to a finite dimensional Hecke algebra with a very nice descrip-tion in terms of generators and relations. In particular this applies to a connectedreductive algebraic group defined over a finite field.On a higher level, if G is a reductive algebraic group over a non-Archimedeanlocal field, then the Hecke algebra H ( G ) has infinite dimension. However, thevalution of the field provides enough extra structure to show that H ( G ) is a directsum of factors which tend to be Morita equivalent to affine Hecke algebras.In the the first section we browse through the literature on reductive groups,and we report when and how we see something that looks like an affine Heckealgebra. Meanwhile we also recall some important notions from the representationtheory of totally disconnected groups (like groups over a p -adic field).In Section 4.2 we start working towards the main new result of this chapter,namely the construction of natural isomorphisms HP ∗ ( H ( G )) ∼ = HP ∗ ( S ( G ) , ⊗ ) ∼ = K ∗ ( C ∗ r ( G )) (4.1)Obviously we have to recall the definitions of the involved algebras. The reduced C ∗ -algebra of G is defined in a standard way, but the construction of the Schwartzalgebra S ( G ), originally due to Harish-Chandra [55], is much more difficult. Thegreater part of Section 4.2 is used to give a proper definition of this algebra, andto characterize its representations among all G -representations.Acknowledging the difference between H ( G ) and affine Hecke algebras, we stillproceed like we did in Chapter 3. Thus for information about the primitive idealspectra of H ( G ) and S ( G ) we turn to the Fourier transform and the Planchereltheorem for reductive p -adic groups, both of which are due to Harish-Chandra[56]. These will show that Prim( H ( G )) is a countable union of non-separatedcomplex affine varieties. The algebras S ( G ) and C ∗ r ( G ) have the same spectrum,which turns out to be a countable union of compact non-Hausdorff spaces. TheLanglands classification tells us that Prim( S ( G )) is in a sense a deformation retractof Prim( H ( G )). 10304 Chapter 4. Reductive p -adic groupsNearly all the new material of this chapter is contained in the final section.There we use all the above to lift the result (3.1) to Hecke algebras of reductive p -adic groups, which yields (4.1). We remark that we are careful with topologicalperiodic cyclic homology, here we have to take it with respect to the completedinductive tensor product ⊗ .Moreover (4.1) is related to the Baum-Connes conjecture for reductive p -adicgroups by means of the diagram K G ∗ ( βG ) → K ∗ ( C ∗ r ( G )) ↓ ↓ HP ∗ ( H ( G )) → HP ∗ ( S ( G ))We conclude the chapter with a discussion of some subtleties of this diagram. In Section 3.1 we defined Iwahori-Hecke algebras in terms of generators and rela-tions, but this is hardly the way in which they emerged. Iwahori and Matsumoto[63, 64, 65, 89] discovered that convolution algebras associated to a reductive groupand a suitable subgroup are of the type we described. We have a look at theseand then we extend our view to more general convolution algebras, of smoothfunctions on reductive p -adic groups. We recall everything that is needed to stateall the known cases in which such convolution algebras yield affine Hecke algebrasor closely related structures.The most direct way to arrive at Iwahori-Hecke algebras is through groups witha BN -pair. Recall that a group G has a BN -pair if it satisfies1. G is generated by two subgroups B and N B ∩ N is normal in N W := N/B ∩ N is generated by a set S = { s i : i ∈ I } of elements of order 24. if n i ∈ N and n i ( B ∩ N ) = s i then n i Bn i (cid:54) = B
5. for all such n i and n ∈ N we have n i Bn ⊂ Bn i nB ∩ BnB
These axioms were first formalized by Tits, cf. [19]. Some important conse-quences are proven in [13, Chapitre IV.2]. For example, it turns out that (
W, S )is a Coxeter system and that the group G has a Bruhat decomposition, i.e. thereis a bijection between W and the double cosets of B in G , given by N (cid:51) n → BnB ∈ B \ G/B (4.2)Any connected reductive group G over an algebraically closed field K has a BN -pair. To be precise, in this case B is a Borel subgroup of G , B ∩ N = T is a.1. Hecke algebras of reductive groups 105maximal torus and N is the normalizer of T in G , see [124, Chapter 8]. With G and T one can associate a root datum R ( G , T ) = ( X, Y, R , R ∨ ) where • X is the character lattice of T• Y is the cocharacter lattice of T• R is the set of roots of ( G , T ) • R ∨ is the set of coroots of ( G , T ) • W = N / T is isomorphic to the Weyl group W of R This results in a bijection between isomorphism classes of connected reductivealgebraic groups and root data, see [30, Expos´e 24] and [38, Expos´e XXV]. Underthis bijection semisimple groups correspond to semisimple root data.The most important example is of course GL ( n, K ). In this group we may takefor B the subgroup of upper triangular matrices and for T the subgroup of diagonalmatrices. Then N consists of the matrices that have exactly one nonzero entry inevery row and every column. In the root datum R ( G , T ) we have X ∼ = Y ∼ = Z n and R = R ∨ = { e i − e j : 1 ≤ i, j ≤ n, i (cid:54) = j } where { e i } ni =1 is the standard basis of Z n . So R is the root system A n − , and theWeyl group W is isomorphic to the symmetric group S n .Assume now that we have a group with a BN -pair such that B is almostnormal in G . This means that every double coset of B is a finite union of leftcosets. (Clearly B would be almost normal if it were normal in G , but this canonly happen in the degenerate situation B = G, N = { e } .) Let k be a unitalcommutative ring and consider the k -module H ( G, B ) of k -valued B -biinvariantfunctions on G that are nonzero on only finitely many left B -cosets. Clearly thisis a free module with basis { T w : w ∈ W } , where T w is the characteristic functionof BwB . (This sloppy notation is justified by (4.2).) Define a measure µ on B -leftinvariant subsets by µ ( H ) = | B \ H | . The product on H ( G, B ) is convolution withrespect to µ : ( f ∗ f )( w ) = (cid:90) B \ G f ( wx − ) f ( x ) dµ ( x ) (4.3)This notion of a Hecke algebra stems from Shimura [117, § Theorem 4.1
For w ∈ W write q ( w ) = µ ( BwB ) = | B \ BwB | Then q is a label function and the relations (3.3) and (3.4) hold in H ( G, B ) .
06 Chapter 4. Reductive p -adic groups Proof.
This result is due to Iwahori [63, Theorem 3.2] and Matsumoto [89,Th´eor`eme 4]. A full proof can be found in [46, Theorem 8.4.6]. (cid:50)
In the above situation of a reductive group G over an algebraically closed field K , B cannot be almost normal, because it has lower dimension than G and K isinfinite. However, suppose that the characteristic p of K is nonzero and that G is defined over a finite field F q . The group G ( F q ) of F q -rational points still has a BN -pair, where B ( F q ) is a Borel subgroup. The associated Hecke algebras werestudied by Iwahori [63] (for Chevalley groups) and by Howlett and Lehrer, see[27]. In these cases W is a subgroup of the finite Weyl group W of R ( G , T ), andthe numbers q ( s ) are certain powers of p .Now we turn to p -adic groups. Recall that a non-Archimedean local field is afield F with a discrete valuation v : F → Z ∪ {∞} (4.4)such that F is complete with respect to the induced norm (cid:107) x (cid:107) F = q − v ( x ) . Here q is the cardinality of the residue field O / P , O = { x ∈ F : v ( x ) ≥ } (4.5)being the ring of integers of F and P = { x ∈ F : v ( x ) > } (4.6)its unique maximal ideal. This implies that F is a totally disconnected, nondiscrete,locally compact Hausdorff space. If its characteristic is zero then F is isomorphicto a finite algebraic extension of the field of p -adic numbers Q p . On the otherhand, if char( F ) > O ∼ = F q [[ t ]], the ring of formal power series over thefinite field F q . With a slight abuse of terminology, non-Archimedean local fieldsare also known as p -adic fields.So let F be a non-Archimedean local field and G a connected reductive algebraicgroup that is defined over F . Consider the group G = G ( F ) of F -rational pointsof G . Affine Hecke algebras play an important role in the representation theoryof such groups, as we will try to explain. We are mainly interested in smoothrepresentations, i.e. representations of G on a complex vector space V such thatfor every v ∈ V the group { g ∈ G : gv = v } is open. Recall that, because F isnon-Archimedean, the identity element e of G has a countable neighborhood basisconsisting of compact open subgroups. Hence smooth representations can also becharacterized by the condition V = (cid:91) K V K where K runs over all compact open subgroups of G . We denote the category ofsmooth G -representations by Rep( G ), and the set of equivalence classes of irre-ducible smooth G -representations by Irr( G ). We call a map from G to a Hausdorff.1. Hecke algebras of reductive groups 107space smooth if it is uniformly locally constant, i.e. if it is bi-invariant for somecompact open subgroup of G .Furthermore, because G is reductive, it has an affine building βG , also known asthe Bruhat-Tits building of G . We quickly recall the construction and terminologyof this polysimplicial complex, referring to [19, 127] for more detailed information.Let A = A ( F ) be a maximal F -split torus of G , X ∗ ( A ) = X ∗ ( A ) its characterlattice, and put a ∗ = X ∗ ( A ) ⊗ Z R . Then βG is G × a ∗ modulo a certain equivalencerelation.The affine building is a universal space for proper G -actions. Such a universalspace always exists, based on general categorical considerations, but it is uniqueonly up to homotopy. On the other hand, the proof that βG really has the requiredproperties is very tricky, and ultimately relies on the existence of “valuated rootdata” [20].The images of a ∗ under G are the appartments of βG , and a polysimplexof maximal dimension in βG is called a chamber. The stabilizer I of such achamber (or equivalently of an interior point of a chamber) is an Iwahori subgroupof G . More generally the stabilizer of an arbitrary point of βG is known as aparahoric subgroup. If x ∈ βG is a “special” point (in particular it must lie ina polysimplex of minimal dimension) then its stabilizer K is a ”good” maximalcompact subgroup of G in the sense of [118, § G = P K = K P (4.7)for any parabolic subgroup P of G containing A .Normalize the Haar measure µ on G by µ ( K ) = 1. Any compact open K < G is almost normal, so we can consider the convolution algebra H ( G, K ) of compactlysupported C -valued K -biinvariant functions on G . For example, if G = GL ( Q p )and K = GL ( Z p ), then H ( G, K ) is the classical algebra of Hecke operators, hencethe ” H ” for ”Hecke” algebra.If K (cid:48) ⊂ K is another compact subgroup then there is a natural inclusion H ( G, K ) → H ( G, K (cid:48) ). The inductive limit of this system of inclusions (i.e. theunion), over all compact open subgroups, is called the Hecke algebra H ( G ) of G .It consists of all compactly supported smooth functions on G . For every K wedefine the idempotent e K ∈ H ( G ) by e K ( g ) = (cid:26) µ ( K ) − if g ∈ K g / ∈ K (4.8)This gives the useful identification e K H ( G ) e K = H ( G ) K × K = H ( G, K ) (4.9)By construction a smooth G -representation is the same thing as a nondegeneraterepresentation of H ( G ). There are natural mapsRep( G ) → Rep (cid:0) H ( G, K ) (cid:1) : V → V K = π ( e K ) V Rep (cid:0) H ( G, K ) (cid:1) → Rep( G ) : W → Ind H ( G ) H ( G,K ) W (4.10)08 Chapter 4. Reductive p -adic groupsBernstein [9, Corollaire 3.9] showed that there exist arbitrarily small K for whichthese maps define equivalences between the category of nondegenerate H ( G, K )-representations and the category of those smooth G -representations that are gen-erated by their K -fixed vectors. Thus H ( G, K ) covers a clear part of the repre-sentation theory of G .It is quite possible that H ( G, K ) is an extended Iwahori-Hecke algebra. Forexample, suppose that G is split over F , let A = A ( F ) be a maximal split torusand B = B ( F ) a Borel subgroup containing T . Assume that B is defined over O ,so that B ( O / P ) is a Borel subgroup of G ( O / P ). The inverse image I of B ( O / P )under the quotient map G ( O ) → G ( O / P ) is an Iwahori subgroup of G . It is known[65] that H ( G, I ) is an affine Hecke algebra. The Weyl group of the associated rootdatum is W = N G ( A ) / A ( O ) and it decomposes as W = W (cid:110) X ∗ ( A ) where W = N G ( O ) ( A ) / A ( O ). Finally, the value of the label function on any simplereflection is q = |O / P | .Or suppose that G is simply connected, but not necessarily split over F . Let N be the stabilizer of an appartment in the affine building of G , and I the stabilizerof a chamber of this appartment. According to [20, Proposition 5.2.10] ( I, N )is a BN -pair in G , so H ( G, I ) is an Iwahori-Hecke algebra. The Coxeter group W = N/I ∩ N is an affine Weyl group coming from a root datum that is containedin R ( G , T ), for a suitable torus T .An important decomposition of the category Rep( G ) was discovered by Bern-stein [9]. To describe it we introduce several classes of smooth representations.We call ( π, V ) ∈ Rep( G ) • admissible if V K has finite dimension for every compact open subgroup K • supercuspidal if it is admissible and all matrix coefficients of π have compactsupport modulo the center of G By [9, Corollaire 3.4] for every K the algebra H ( G, K ) is of finite type, so all itsirreducible representations have finite dimension. In combination with [9, Corol-laire 3.9 ] this shows that every irreducible smooth representation is automaticallyadmissible. Supercuspidal representations are also known as absolutely cuspidal(or just cuspidal) representations, but there seems to be no agreement in the ter-minology here.There is a natural notion of the contragredient of a smooth representation. Let˘ V K = { f ∈ V ∗ : f ◦ π ( e K ) = f } (4.11)be the dual space of V K and define˘ V = (cid:91) K ˘ V K (4.12)Then (˘ π, ˘ V ) is the contragredient representation of ( π, V ). By construction it issmooth, and it is admissible whenever V is..1. Hecke algebras of reductive groups 109Suppose that P is a parabolic subgroup of G and that M is a Levi subgroupof P . Although G and M are unimodular, the modular function δ P of P is ingeneral not constant. Let σ be an irreducible supercuspidal representation of M .Under these conditions we call ( M, σ ) a cuspidal pair. From this we construct aparabolically induced G -representation I GP ( σ ) = Ind GP (cid:0) δ / P ⊗ σ (cid:1) (4.13)This means that we first inflate σ to P , and then we apply the normalized inductionfunctor, i.e. we twist it by δ / P and take the smooth induction to G . This twist isuseful to preserve unitarity, cf. [28, Theorem 3.2].For every ( π, V ) ∈ Irr( G ) there is a cuspidal pair ( M, σ ), uniquely determinedup to G -conjugacy, such that V is a subquotient of I GP ( σ ). If P (cid:48) is another parabolicsubgroup of G containing M then I GP (cid:48) ( σ ) and I GP ( σ ) have the same irreduciblesubquotients, but they need not be isomorphic.To define a suitable equivalence relation on the set of cuspidal pairs, we nowrecall a particular kind of characters. Let H be any algebraic group over F , andconsider the subgroup H = { h ∈ H : v ( γ ( h )) = 0 ∀ γ ∈ X ∗ ( H ) } (4.14)This is an open normal subgroup of G which contains every compact subgroup, and H/ H is a free abelian group. An unramified character of H is a homomorphism χ : H → C × whose kernel contains H . The group of these forms a complex torus X nr ( H ) and the map X ∗ ( H ) ⊗ Z C × → X nr ( H ) defined by γ ⊗ z −→ (cid:16) h → zv ( γ ( h )) (cid:17) is an isomorphism. We will denote the compact torus of unitary unramified char-acters by X unr ( H ) = Hom (cid:0) H/ H, S (cid:1) (4.15)We say that two cuspidal pairs ( M, σ ) and ( M (cid:48) , σ (cid:48) ) are inertially equivalent if thereexist χ ∈ X nr ( M (cid:48) ) and g ∈ G such that M (cid:48) = g − M g and σ (cid:48) ⊗ χ ∼ = σ g .With an inertial equivalence class s = [ M, σ ] G we associate a subcategory ofRep( G ) s of Rep( G ). By definition its objects are those smooth representations π with the following property: for every irreducible subquotient ρ of π there is a( M, σ ) ∈ s such that ρ is a subrepresentation of I GP ( σ ).These Rep( G ) s give rise to the Bernstein decomposition [9, Proposition 2.10]:Rep( G ) = (cid:89) s ∈ B ( G ) Rep( G ) s (4.16)The set B ( G ) of Bernstein components is countably infinite. We have a corre-sponding decomposition of the Hecke algebra of G into two-sided ideals: H ( G ) = (cid:77) s ∈ B ( G ) H ( G ) s (4.17)10 Chapter 4. Reductive p -adic groupswith Rep (cid:0) H ( G ) s (cid:1) = Rep( G ) s . By [24, Proposition 3.3] there exists an idempotent e s ∈ H ( G ) such that • H ( G ) s = H ( G ) e s H ( G ) • Rep( G ) s is equivalent to Rep (cid:0) e s H ( G ) e s (cid:1) Under these conditions e s H ( G ) e s is a finite type algebra, whose center was alreadydescribed by Bernstein. The set D σ of all cuspidal pairs of the form ( M, σ ⊗ χ ) isin bijection with X nr ( M ), so it has the structure of a complex torus. Put N ( M, σ ) = (cid:8) g ∈ G : gM g − = M and [ M, σ g ] M = [ M, σ ] M (cid:9) (4.18)Then W σ = N ( M, σ ) /M is a finite group acting on D σ , so D σ /W σ is an irreduciblealgebraic variety. [9, Th´eor`eme 2.13] tells us that Z (cid:0) e s H ( G ) e s (cid:1) ∼ = O ( D σ /W σ ) = O ( D σ ) W σ (4.19)For any compact open K < G write B ( G, K ) = (cid:8) s ∈ B ( G ) : H ( G, K ) s (cid:54) = 0 (cid:9) H ( G, K ) s = H ( G ) s ∩ H ( G, K ) Proposition 4.2 B ( G, K ) is finite for any compact open K < G .2. If K is a normal subgroup of a good maximal compact subgroup K then H ( G, K ) is Morita equivalent to ⊕ s ∈ B ( G,K ) H ( G ) s and Z (cid:0) H ( G, K ) (cid:1) ∼ = (cid:77) s ∈ B ( G,K ) Z (cid:0) e s H ( G ) e s (cid:1)
3. For any s ∈ B ( G ) there exists a compact open K s < G such that for everycompact open K ⊂ K s the algebras H ( G, K s ) s , H ( G, K ) s and H ( G ) s are Morita equivalent.Proof. All these results are due to Bernstein. 2 is a direct consequence of [9,Corollaire 3.9] and (4.19). 1 and 3 follow from this in combination with [9, (3.7)],as was remarked in [7, p. 143]. (cid:50)
We may also consider more general algebras associated to (
G, K ). Let ( ρ, V ) bean irreducible smooth representation of K , and (˘ ρ, ˘ V ) its contragredient. Noticethat ρ is smooth and has finite dimension. Define H ( G, K, ρ ) = (cid:110) f : G → End C ( ˘ V ) : f ( k gk ) = ˘ ρ ( k ) f ( g )˘ ρ ( k ) ∀ k , k ∈ K, g ∈ G (cid:111) (4.20).2. Harish-Chandra’s Schwartz algebra 111This is a unital algebra under the convolution product, its elements being smoothfunctions on G . Consider the idempotent e ρ ∈ H ( G ) defined by e ρ ( g ) = (cid:26) µ ( K ) − dim( V ) tr (cid:0) ρ ( g − ) (cid:1) if g ∈ K g / ∈ K (4.21)By [23, Proposition 4.2.4] there is a natural isomorphism H ( G, K, ρ ) ⊗ C End C ( V ) ∼ = e ρ H ( G ) e ρ (4.22)Let Rep ρ ( G ) be the subcategory of Rep( G ) consisting of all representations ( π, U )for which H ( G ) e ρ U = U . According to [23, Proposition 4.2.3] there are naturalbijections between the sets of irreducible objects of • Rep ρ ( G ) • Rep (cid:0) e ρ H ( G ) e ρ (cid:1) • Rep (cid:0) H ( G, K, ρ ) (cid:1) If moreover Rep ρ ( G ) is closed under taking subquotients (of G -representations)then there exists a finite subset S ⊂ B ( G ) such thatRep ρ ( G ) = (cid:77) s ∈ S Rep( G ) s (4.23)In the terminology of Bushnell and Kutzko ( K, ρ ) is an S -type [24, (3.12)]. Ofspecial interest is the case when S consists of a single element s , for then H ( G, K, ρ )is Morita equivalent to H ( G ) s . It is known that under this and certain extraconditions H ( G, K, ρ ) is isomorphic to an affine Hecke algebra [108, Theorem 6.3],sometimes with unequal labels [87, Section 1], or to the twisted crossed productof such a thing with a (twisted) group algebra [94, Theorem 7.12].Using this approach it has been shown that H ( G ) s is Morita equivalent to anaffine Hecke algebra for every s ∈ B ( GL ( n, F )) [23, 25], and to a “twisted” affineHecke algebra for every s ∈ B ( SL ( n, F )) [48, Theorem 11.1]. In this section G will be a connected reductive algebraic group defined over a non-Archimedean local field F . The reduced C ∗ -algebra of G is defined in a standardway, but we need to go to some lengths to construct Harish-Chandra’s Schwartzalgebra. Once this is done we characterize its representations among admissible G -representations and formulate the Langlands classification for reductive p -adicgroups.12 Chapter 4. Reductive p -adic groupsDefine the adjoint and the trace of f ∈ H ( G ) by f ∗ ( g ) = f ( g − ) τ ( f ) = f ( e )This gives rise to a bitrace ( f, f (cid:48) ) = τ ( f ∗ f (cid:48) )making H ( G ) into a Hilbert algebra. The Hilbert space completion of H ( G ) isthe space L ( G ) of all square-integrable functions on G . It carries two natural G -actions, left and right translation: (cid:0) λ ( g ) f (cid:1) ( h ) = f ( g − h ) (cid:0) ρ ( g ) f (cid:1) ( h ) = f ( hg )Now λ ( g ) and ρ ( g ) are bounded operators of the same norm. They extend naturallyto representations of H ( G ), so we get an injection λ : H ( G ) → B ( L ( G ))The reduced C ∗ -algebra C ∗ r ( G ) is the closure of λ ( H ( G )) in B ( L ( G )). It isa separable nonunital C ∗ -algebra whose representations correspond to the uni-tary G -representations that are weakly contained in the left regular representation( λ, L ( G )) of G .Usually the reduced C ∗ -algebra of a locally compact group H is defined as thecompletion of C c ( H ) or L ( H ) in B ( L ( H )). However if H is totally disconnectedwe may just as well start with smooth functions only.For a compact open K < G we let C ∗ r ( G, K ) be the completion of H ( G, K ) in B ( L ( G )). It is a unital type I C ∗ -algebra and it equals e K C ∗ r ( G ) e K = C ∗ r ( G ) K × K = C ∗ r ( G, K ) (4.24)Let us mention some general facts about the structure of C ∗ r ( G ). They can be readoff from Theorem 4.10, but it seems appropriate to formulate them here already.This algebra can be recovered as the inductive limit of the above subalgebras, overall compact open subgroups, partially ordered by inclusion: C ∗ r ( G ) = lim −→ C ∗ r ( G, K ) (4.25)Moreover it has a Bernstein decomposition, analogous to (4.17), with a direct sumin the C ∗ -algebra sense: C ∗ r ( G ) = lim −→ S (cid:77) s ∈ S C ∗ r ( G ) s (4.26)where S runs over all finite subsets of B ( G ). Here C ∗ r ( G ) s is the two-sided idealof C ∗ r ( G ) generated by H ( G ) s . Every subalgebra C ∗ r ( G, K ) lives in only finitelymany Bernstein components..2. Harish-Chandra’s Schwartz algebra 113The construction of the Schwartz algebra of G is more complicated, we needto introduce a lot of things to achieve it.A p-pair is a pair ( P, A ) consisting of a parabolic subgroup P of G , and theidentity component A of the maximal split torus in the center of some Levi sub-group M of P . Then M = Z G ( A ) = A × M (4.27) P = M N = A M N = Z G ( A ) N (4.28)where N is the unipotent radical of P . For example ( G, A G ) is a p-pair, where A G is the maximal split torus of Z ( G ).There is a unique p-pair ( ¯ P , A ) such that ¯ P ∩ P = M . The parabolic subgroup¯ P is called the opposite of P . Clearly ¯ P = M ¯ N where ¯ N ∩ N = { } .Let ( Q, B ) be another parabolic pair. Write W ( A | G | B ) for the set of all ho-momorphisms B → A induced by inner automorphisms of G . If B = A then thisis a group : W ( G | A ) := W ( A | G | A ) = N G ( A ) /Z G ( A ) = N G ( A ) /M (4.29)We say that ( P, A ) dominates (
Q, B ), written (
P, A ) ≥ ( Q, B ), if P ⊃ Q and A ⊂ B .Recall that we have chosen a maximal split torus A of G , and let P be aminimal parabolic subgroup containing it. We call a p-pair ( P, A ) and its Levifactor M semi-standard if A ⊂ A , or equivalently A ⊂ M . If moreover( P, A ) ≥ ( P , A ), then we say that ( P, A ) is standard. Every p-pair is conjugateto a standard p-pair.Let X ∗ ( A ) be the character lattice of A and put a = Hom Z ( X ∗ ( A ) , R ) a ∗ = X ∗ ( A ) ⊗ Z R (4.30)There is a natural homomorphism H M : M → a , defined by the equivalent condi-tions (cid:104) χ , H M ( m ) (cid:105) = − v ( χ ( m )) q (cid:104) χ , H M ( m ) (cid:105) = (cid:107) χ ( m ) (cid:107) F (4.31)where χ ∈ X ∗ ( A ) and q is the module of F . Conversely, if ν ∈ a ∗ then we definean unramified character χ ν of M by χ ν ( m ) = q (cid:104) ν , H M ( m ) (cid:105) (4.32)For a parabolic subgroup Q with P ⊂ Q ⊂ G , let Σ( Q, A ) ⊂ a ∗ be the set of rootsof Q with respect to A . By this we mean the set of α ∈ X ∗ ( A ) \ { } such that q α is nonzero, where q is the Lie algebra of Q and q α := { x ∈ q : Ad( a ) x = α ( a ) x ∀ a ∈ A }
14 Chapter 4. Reductive p -adic groupsThen Σ( G, A ) is a root system and Σ(
P, A ) is a positive system of roots. Let∆(
P, A ) be the corresponding set of simple roots. The Weyl group of Σ(
G, A ) isnaturally isomorphic to W ( G | A ). Notice that Σ( ¯ P , A ) = − Σ( P, A ).The minimal p-pair ( P , A ) gives us a root systemΣ = Σ( G, A ) ⊂ a ∗ with simple roots ∆ = ∆( P , A ) and Weyl group W = W ( G | A ). Fix a W -invariant inner product on a ∗ , so that we may identify this vector space with itsdual a .If ( P, A ) is standard then ∆(
P, A ) is the set of nonzero projections of ∆ on a ∗ , and W ( M | A ) is the parabolic subgroup of W generated by { s α : α ∈ ∆ , α ( A ) = 1 } Let us also introduce the associated sets of positive elements in a ∗ and A : a ∗ , + = { ν ∈ a ∗ : (cid:104) ν , α (cid:105) > ∀ α ∈ ∆( P, A ) } ¯ a ∗ , + = { ν ∈ a ∗ : (cid:104) ν , α (cid:105) ≥ ∀ α ∈ ∆( P, A ) } A + = { a ∈ A : (cid:107) α ( a ) (cid:107) F > ∀ α ∈ ∆( P, A ) } ¯ A + = { a ∈ A : (cid:107) α ( a ) (cid:107) F ≥ ∀ α ∈ ∆( P, A ) } In order to say when a function on G is rapidly decreasing, we need a lengthfunction on this group. For x ∈ GL ( m, F ) let x ij and x ij be the entries of x and x − , and define N ( x ) = max {− v ( x ij ) , − v ( x ij ) : 1 ≤ i, j ≤ m } (4.33)Notice that for all x, y ∈ GL ( m, F )0 ≤ N ( xy ) ≤ N ( x ) + N ( y ) (4.34)Pick an injective homomorphism τ : G → GL ( m, F ) and put σ = N ◦ γ : G → Z ≥ (4.35)Then σ is a continuous length function on G . Let δ P be the modular function of P . Using the decomposition (4.7) we extend this to a function δ on G satisfying δ ( pk ) = δ P ( p ) p ∈ P , k ∈ K (4.36)Harish-Chandra’s spherical Ξ-function isΞ( g ) = (cid:90) K δ ( kg ) dµ ( k ) (4.37)Important properties of this function can be found in [132, Paragraphe II] and[118, § n ∈ N consider the following norm on H ( G ) : ν n ( f ) = sup g ∈ G | f ( g ) | Ξ( g ) − ( σ ( g ) + 1) n (4.38).2. Harish-Chandra’s Schwartz algebra 115We say that f ∈ C ( G ) decreases rapidly if ν n ( f ) < ∞ ∀ ∈ N . Clearly the ν n depend on the choice of τ : G → GL ( m, F ), but the topology defined by the family { ν n } ∞ n =1 does not. For any compact open K < G let S ( G, K ) be the completion of H ( G, K ) for this family of norms. According to Vign´eras [130, Theorem 29] this isa unital, nuclear Fr´echet *-algebra, and a dense subalgebra of C ∗ r ( G, K ). Moreoveran element of S ( G, K ) is invertible if and only if it is invertible in C ∗ r ( G, K ), so S ( G, K ) is closed under the holomorphic functional calculus of C ∗ r ( G, K ).For K (cid:48) ⊂ K there is still an inclusion S ( G, K ) → S ( G, K (cid:48) ), so we can takethe inductive limit over all compact open subgroups of G . This yields Harish-Chandra’s Schwartz algebra: S ( G ) = lim −→ S ( G, K ) (4.39)By definition it consists of all rapidly decreasing smooth functions on G . Theobvious analogue of (4.9) is e K S ( G ) e K = S ( G ) K × K = S ( G, K ) (4.40)Compared to the above C ∗ -algebras, S ( G ) inherits fewer topological propertiesfrom its subalgebras. Namely, it is a complete Hausdorff locally convex algebra,but it is not metrizable, and its multiplication is only separately continuous [132, § III.6].It does have a Bernstein decomposition S ( G ) = (cid:77) s ∈ B ( G ) S ( G ) s (4.41)where S ( G ) s is the completion of H ( G ) s , a two-sided ideal in S ( G ). This followsfrom Theorem 4.9, but of course it can be proved without using the full strengthof that result.To characterize those G -representations that extend to S ( G ) we need to knowmore about smooth representations.Let ( π, V ) be a smooth G -representation, and P a parabolic subgroup withunipotent radical N and a Levi factor M . The Jacquet module associated to thesedata is V P = V /V ( N ) V ( N ) = span { π ( n ) v − v : n ∈ N, v ∈ V } (4.42)We make it into an M -representation ( π P , V P ) by π P ( m ) j P ( v ) = δ − / P ( m ) j P ( π ( m ) v ) (4.43)where j P : V → V /V ( N ) is the natural projection. By Frobenius reciprocity weget, for any smooth M -representation σ :Hom G ( π, I GP σ ) ∼ = Hom M ( π P , σ ) (4.44)16 Chapter 4. Reductive p -adic groupsFor χ ∈ Hom( A G , C × ) define the generalized weight space V χ = (cid:8) v ∈ V : ∃ n ∈ N : ( π ( a ) − χ ( a )) n v = 0 ∀ a ∈ A G (cid:9) (4.45)If V χ (cid:54) = 0 then we call χ an exponent of ( π, V ). If π ( a ) v = χ ( a ) v ∀ v ∈ V, a ∈ A G then we say that V admits the central character χ . Similarly for a p-pair ( P, A )and a smooth M -representation we have exponents in Hom( A, C × ). The set ofexponents of the Jacquet module V ¯ P is X π ( P, A ) = { χ ∈ Hom( A, C × ) : V ¯ P ,χ (cid:54) = 0 } (4.46)Notice the curious shift in the notation, from ¯ P to P . This is designed to make anicer formulation of the Langlands classification possible.Let us characterize square-integrable, discrete series and tempered G -representations. Our first description is due to Casselman [29, Theorem 4.4.6]. Proposition 4.3
Let π be an admissible G -representation which admits a unitarycentral character. The following are equivalent : • Every matrix coefficient of π is square-integrable on G/A G • If ( P, A ) is a semi-standard p-pair, χ ∈ X π ( P, A ) and a ∈ ¯ A + is such thatthere is an α ∈ ∆ with | α ( a ) | (cid:54) = 1 , then | χ ( a ) | < • For every semi-standard p-pair ( P, A ) and every χ ∈ X π ( P, A ) we can write log | χ | = (cid:88) α ∈ ∆( P,A ) χ α α with χ α < We say that π is square-integrable if it satisfies these conditions. Every square-integrable representation is unitary and completely reducible, see[118, Corollary 1.11.8] or [132, Lemme III.1.3]. A more restrictive notion is thatof a discrete series representation.
Proposition 4.4
Let ( π, V ) be an irreducible admissible G -representation. Thefollowing are equivalent : • ( π, V ) is a subrepresentation of ( λ, L ( G )) • G is semi-simple and π is square-integrableIf ( π, V ) satisfies these conditions then it is called a discrete series representation. .2. Harish-Chandra’s Schwartz algebra 117By [41, Proposition 18.4.2] such a representation does indeed give an isolatedpoint in Prim( C ∗ r ( G )).A (smooth) function f on G is tempered if there exist C, N ∈ (0 , ∞ ) such that | f ( g ) | ≤ C Ξ( g )(1 + σ ( g )) N ∀ g ∈ G (4.47) Proposition 4.5
Let π be an admissible G -representation. The following areequivalent : • π extends continuously to S ( G ) • Every matrix coefficient of π is a tempered function • If ( P, A ) is a semi-standard p-pair, χ ∈ X π ( P, A ) and a ∈ A + , then | χ ( a ) | ≤ • For every semi-standard p-pair ( P, A ) and every χ ∈ X π ( P, A ) we can write log | χ | = (cid:88) α ∈ ∆( P,A ) χ α α with χ α ≤ The representation π is said to be tempered if and only if these conditions hold.Proof. Almost everything follows from [132, Proposition III.2.2 and § III.7]. Theonly thing left is to show that all matrix coefficients of an admissible S ( G )-representation are tempered. This follows from the admissibility, combined withthe observation that the collection of tempered K -biinvariant functions on G isthe linear dual of S ( G, K ) . (cid:50) The properties temperedness and pre-unitarity are preserved under normalizedinduction:
Proposition 4.6
Let ( P, A ) be a semi-standard p-pair and ( π, V ) an admissible M -representation. Then1. I GP ( π ) is tempered if and only if π is tempered2. I GP ( π ) is pre-unitary if and only if π is pre-unitaryProof.
1. comes from [132, Lemme III.2.3].2. It is clear that I GP ( π ) cannot be pre-unitary if π is not. It remains to producea G -invariant inner product on I GP ( V ), given an M -invariant inner product on V .This is achieved by setting (cid:104) f , f (cid:48) (cid:105) = (cid:90) K (cid:104) f ( k ) , f (cid:48) ( k ) (cid:105) dµ ( k ) (4.48)18 Chapter 4. Reductive p -adic groupsNotice that I GP ( V ) is usually not complete with respect to this inner product, evenis V is. (cid:50) Like in Section 3.2, let Λ be the set of triples (
P, σ, ν ), where (
P, A ) is a standardp-pair, σ is an irreducible tempered representation of M = Z G ( A ) and ν ∈ a ∗ .With such a triple we associate the admissible G -representation I ( P, σ, ν ) = I GP ( σ ⊗ χ ν ) = Ind GP (cid:0) σ ⊗ χ ν ⊗ δ / P (cid:1) (4.49)The set of Langlands data isΛ + = (cid:8) ( P, σ, ν ) ∈ Λ : ν ∈ a ∗ , + (cid:9) (4.50)A somewhat extended version of the Langlands classification for reductive p -adicgroups reads : Theorem 4.7
Let ( P, σ, ν ) , ( P (cid:48) , σ (cid:48) , ν (cid:48) ) ∈ Λ + .1. The G -representation I ( P, σ, ν ) is indecomposable and has a unique irre-ducible quotient, which we call J ( P, σ, ν ) .2. For every π ∈ Irr( G ) there is precisely one Langlands datum ( P, σ, ν ) suchthat π is equivalent to J ( P, σ, ν ) .3. If J ( P, σ, ν ) is equivalent to a subquotient of I ( P (cid:48) , σ (cid:48) , ν (cid:48) ) , then ν (cid:48) − ν ∈ ¯ a ∗ , +0 and P (cid:48) ⊂ P . If P (cid:48) = P then also σ (cid:48) = σ and ν (cid:48) = ν .Proof. For 1 and 2 see [119] or [11, § XI.2]. As concerns 3, by [11, Lemma XI.2.13]we have ν (cid:48) − ν ∈ ¯ a ∗ , +0 . Now it follows from the definition of Λ + that P (cid:48) ⊂ P .Suppose that ( P, σ, ν ) (cid:54) = ( P (cid:48) , σ (cid:48) , ν (cid:48) ) while P = P (cid:48) . Then, again by [11, LemmaXI.2.13], ν (cid:54) = ν (cid:48) . But by Frobenius reciprocity σ is equivalent to a subquotient of I GP ( σ (cid:48) ⊗ χ ν (cid:48) )( N ). Hence ν = ν (cid:48) ◦ w for some w ∈ W ( G | A ) \ { } . Since ν (cid:48) ∈ a (cid:48)∗ , + there is an α ∈ ∆ with (cid:104) ν (cid:48) , α (cid:105) > (cid:104) ν , α (cid:105) <
0. This contradicts the positivityof ν with respect to P (cid:48) = P. (cid:50) The Plancherel formula for G is an explicit decomposition of the trace τ in termsof the traces of irreducible G -representations. Closely related is the Planchereltheorem, which describes the image of S ( G ) under the Fourier transform. Thecrucial theorems in this section are due to Harish-Chandra [56], but unfortunatelyhe never published the proofs. Based upon Harish-Chandra’s notes, Waldspurger[132] provided full proofs of these results, which we will describe in as much detailas we need. We try to set up a complete analogy with Section 3.3. In particularwe refine the Langlands classification using parabolic induction in stages..3. The Plancherel theorem 119We start with a semi-standard p-pair ( P, A ) and an irreducible square-integrable M -representation ( ω, E ). Let (˘ ω, ˘ E ) be its contragredient, and construct the ad-missible G × G -representation L ( ω, P ) = I G × GP × P ( E ⊗ ˘ E ) = I GP ( E ) ⊗ I GP ( ˘ E ) (4.51)Using (4.48) we make I GP ( E ) K into a finite dimensional Hilbert space, for everycompact open K < G . This allows us to identify ˘ I GP ( E ) with I GP ( ˘ E ) as representa-tions, and with I GP ( E ) as vector spaces. Thus we can turn L ( ω, P ) into a nonunital*-algebra with ( f ⊗ f )( f ⊗ f ) = (cid:104) f , f (cid:105) f ⊗ f (4.52)( f ⊗ f ) ∗ = f ⊗ f (4.53)Notice that for every χ ∈ X unr ( M ) the representation χ ⊗ ω is still square-integrable, and that L ( χ ⊗ ω, P ) can be identified with L ( ω, P ). Let K ω be theset of k ∈ X nr ( M ) such that k ⊗ ω is equivalent to ω . This is a finite subgroup of X unr ( M ). For every k ∈ K ω we pick a unitary intertwiner˜ ω k : ( k ⊗ ω, E ) → ( ω, E ) (4.54)This induces an automorphism of L ( ω, P ) by I ( k, ω ) = I G × GP × P (˜ ω k ⊗ ˜ ω − tk ) = I GP (˜ ω k ) ⊗ I GP (˜ ω − tk ) (4.55)where ˜ ω − tk is the inverse transpose of ˜ ω k . Then I ( k, ω ) ∈ Aut G × G (cid:0) L ( ω, P ) (cid:1) isindependent of the choice of ˜ ω k , and in general nontrivial, cf. [132, § VI.1].It is more difficult to define intertwiners corresponding to elements of the var-ious Weyl groups. First we notice that for any p-pair ( P (cid:48) , A ) with the same Levifactor M, ω can also be lifted to a representation of P (cid:48) that is trivial on N (cid:48) . Let( Q, A g ), with g ∈ G , be yet another semi-standard p-pair, and put n = [ g ] ∈ W ( A g | G | A )The equivalence class of the M g representation ( ω g − , E ) depends only on n , andis therefore denoted by nω .In [132, Paragraphe V] certain normalized intertwiners o c Q | P ( n, ω ) are con-structed. Preferring the simpler notation I ( n, ω ), we recall their properties. Theorem 4.8
Let ( P, A ) , ( P (cid:48) , A (cid:48) ) and ( Q, B ) be semi-standard p-pairs, and n ∈ W ( B | G | A ) . There exists an intertwiner I ( n, χ ⊗ ω ) ∈ Hom G × G ( L ( ω, P ) , L ( nω, Q )) with the following properties : • χ → I ( n, χ ⊗ ω ) is a rational function on X nr ( M )20 Chapter 4. Reductive p -adic groups • I ( n, χ ⊗ ω ) is unitary and regular for χ ∈ X unr ( M ) . • If n (cid:48) ∈ W ( A (cid:48) | G | B ) then I ( n (cid:48) , n ( χ ⊗ ω )) I ( n, χ ⊗ ω ) = I ( n (cid:48) n, χ ⊗ ω )Let Γ rr (cid:0) X nr ( M ); L ( ω, P ) (cid:1) be the algebra of rational sections that are regularon X unr ( M ). We define an action of K ω on this algebra by kf ( χ ) = I ( k, ω ) f ( k − χ ) (4.56)Similarly, for n as in Theorem 4.8 we define an algebra homomorphism n : Γ rr (cid:0) X nr ( M )); L ( ω, P ) (cid:1) → Γ rr (cid:0) X nr ( Z G ( B )); L ( nω, Q ) (cid:1) nf ( χ ) = I ( n, ω ) f ( χ ◦ n ) (4.57)To define the Fourier transform we construct a scheme containing all tempered G -representations. For every Levi subgroup M of G choose a set ∆ M of ir-reducible square-integrable M -representations, with the property that for everysquare-integrable π ∈ Irr( M ) there exists exactly one ω ∈ ∆ M such that π isequivalent to χ ⊗ ω , for some χ ∈ X nr ( M ).Let Ξ be the scheme consisting of all quadruples ( P, A, ω, χ ), with (
P, A ) asemi-standard p-pair, ω ∈ ∆ M and χ ∈ X nr ( M ). This is a countable disjointunion of complex algebraic tori. Let Ξ u be the smooth submanifold obtained bythe restriction χ ∈ X unr ( M ). Notice that Ξ is naturally a finite cover of the set Θdefined in [132, p. 305]. For ξ = ( P, A, ω, χ ) ∈ Ξ we put π ( ξ ) = I GP ( χ ⊗ ω ). Let L Ξ be the vector bundle over Ξ which is trivial on every component and whose fiberat ξ is L ( ω, P ). We say that a section of this bundle is algebraic or rational if itis supported on only finitely many components, and has the required property onevery component. Now we define the Fourier transform F : H ( G ) → O (Ξ; L Ξ ) F ( f )( P, A, ω, χ ) = I ( P, A, ω, χ )( f ) ∈ L ( ω, P ) (4.58)This is not the same as ˆ f ( χ ⊗ ω, P ), as in [132, § VII.1]! We adjusted the latter tomake F multiplicative. Fortunately the difference is not too big, so most resultsremain valid.We construct a locally finite groupoid W as follows. The objects of W aretriples ( P, A, ω ) with (
P, A ) a semi-standard p-pair and ω ∈ ∆ M . The morphismsfrom ( Q, B, η ) to (
P, A, ω ) are pairs ( k, n ) with the following properties • k ∈ K ω • n ∈ W ( A | G | B ) and nB = A • nη is equivalent to χ ⊗ ω , for some χ ∈ X nr ( M ).3. The Plancherel theorem 121The multiplication in W , if possible, is given by( k, n )( k (cid:48) , n (cid:48) ) = ( k ( k (cid:48) ◦ n ) , nn (cid:48) ) (4.59)Let Γ rr (Ξ; L Ξ ) be the algebra of rational sections of L Ξ that are regular on Ξ u :Γ rr (Ξ; L Ξ ) = (cid:77) ( P,A,ω ) Γ rr (cid:0) X nr ( M ); L ( ω, P ) (cid:1) = (cid:77) ( P,A,ω ) (cid:8) f ∈ Q ( O ( X nr ( M ))) ⊗ L ( ω, P ) : f is regular on X unr ( M ) (cid:9) (4.60)From (4.56) and (4.57) we get an action of the groupoid W on this algebra. By con-struction the image of H ( G ) under the Fourier transform consists of W -invariantsections.Because ( ω, E ) is admissible C ∞ ( X unr ( M )) ⊗ L ( ω, P ) K × K ∼ = C ∞ ( X unr ( M )) ⊗ I GP ( E ) K ⊗ I GP ( ˘ E ) K (4.61)is in a natural way a Fr´echet space, for every compact open K < G . Endow C ∞ ( X unr ( M )) ⊗ L ( ω, P ) with the inductive limit topology. This also gives atopology on C ∞ c (Ξ u ; L Ξ ), as the inductive limit of finite direct sums of such al-gebras. Notice that these are all *-algebras by (4.63). Clearly the action of W extends continuously to C ∞ c (Ξ u ; L Ξ ).Now the Plancherel theorem for reductive p -adic groups [132, p. 320] tells usthat Theorem 4.9
The Fourier transform F : S ( G ) → C ∞ c (Ξ u ; L Ξ ) W is an isomorphism of topological *-algebras. This guides us to the Fourier transform of C ∗ r ( G ). For ( ω, E ) as on page 119,let K ( ω, P ) be the algebra of compact operators on the Hilbert space completionof I GP ( E ). Notice that K ( ω, P ) = lim −→ L ( ω, P ) K × K (4.62)in the C ∗ -algebra sense, and that the intertwiner I ( n, ω ) extends to K ( ω, P ) be-cause it is unitary. Let K Ξ be the vector bundle over Ξ whose fiber at ( P, A, ω, χ )is K ( ω, P ), and C (Ξ u ; K Ξ ) the C ∗ -completion of (cid:77) ( P,A,ω ) C ( X unr ( M ); K ( ω, P ))Plymen [104, Theorem 2.5] proved that Theorem 4.10
The Fourier transform extends to an isomorphism of C ∗ -algebras C ∗ r ( G ) ∼ −−→ C (Ξ u ; K Ξ ) W
22 Chapter 4. Reductive p -adic groupsThe subalgebras S ( G, K ) are more manageable than S ( G ), so it pays off todescribe their images under F . Theorem 4.11
Let K be a compact open subgroup of G . There exists a finiteset of triples ( P i , A i , ω i ) , i = 1 , . . . , n K , such that the Fourier transform inducesalgebra homomorphisms H ( G, K ) → (cid:76) n K i =1 (cid:0) O ( X nr ( M i )) ⊗ L ( ω i , P i ) K × K (cid:1) W i S ( G, K ) → (cid:76) n K i =1 (cid:0) C ∞ ( X unr ( M i )) ⊗ L ( ω i , P i ) K × K (cid:1) W i C ∗ r ( G, K ) → (cid:76) n K i =1 (cid:0) C ( X unr ( M i )) ⊗ L ( ω i , P i ) K × K (cid:1) W i where W i is the isotropy group of ( P i , A i , ω i ) in W . The first map is injective, thesecond is an isomorphism of Fr´echet *-algebras and the third is an isomorphismof C ∗ -algebras. For every w ∈ W i there is a rational, unitary element u w ∈ C ∞ ( X unr ( M i )) ⊗ L ( ω i , P i ) K × K such that for every f ∈ C ( X unr ( M i )) ⊗ L ( ω i , P i ) K × K wf ( χ ) = u w ( χ ) f ( w − χ ) u − w ( χ ) (4.63) Proof.
By [132, Th´eor`eme VIII.1.2] there is only a finite number of associationclasses among the objects of W on which the idempotent e K does not act as zero.Pick one representant ( P i , A i , ω i ) in every such association class. From (4.58) andTheorems 4.9 and 4.10 we immediately get the required description of the Fouriertransforms of H ( G, K ) , S ( G, K ) and C ∗ r ( G, K ).Every automorphism of L ( ω i , P i ) K × K ∼ = End (cid:0) I GP ( E ) K (cid:1) is inner, so the formula(4.63) holds for some u w . Using Theorem 4.8 we can arrange that u w is rationalon X nr ( M i ) and unitary on X unr ( M i ) . (cid:50) Purely representation-theoretic consequences of the above isomorphisms are:
Corollary 4.12
1. Every irreducible tempered G -representation is a direct sum-mand of I ( ξ ) , for some ξ ∈ Ξ u .2. For any w ∈ W and ξ ∈ Ξ such that wξ is defined, the G -representations I ( ξ ) and I ( wξ ) have the same irreducible subquotients, counted with multiplicity.Proof.
1. Let K be a compact open subgroup of G such that V K (cid:54) = 0, and let V (cid:48) be an irreducible submodule of V K , considered as a S ( G, K )-representation.With Theorem 4.11 and the same argument as in the proof of Corollary 3.26.2 wededuce that V (cid:48) is a direct summand of I ( P i , A i , ω i , χ ) K for some χ ∈ X unr ( M ).Because V (cid:48) generates V as a G -module, V is a constituent of I ( P i , A i , ω i , χ ). ByProposition 4.6.2 the latter is completely reducible, so V is in fact equivalent to adirect summand.of I ( P i , A i , ω i , χ )..3. The Plancherel theorem 1232. By [29, Corollary 2.3.3] we have to show that the characters of I ( ξ ) and I ( wξ )are the same, i.e. that the function H ( G ) × X nr ( M ) → C : ( f, χ ) → tr I ( P, A, ω, χ )( f ) − tr I ( wP, wA, wω, χ ◦ w − )( f )(4.64)is identically 0. Because this is a polynomial function of χ , it suffices to show thatit is 0 on H ( G ) × X unr ( M ). This is an immediate consequence of Theorem 4.9. (cid:50) For ξ = ( P, A, ω, χ ) ∈ Ξ we define A ( ξ ) = { a ∈ A : α ( a ) = 1 if (cid:104) log | χ | , α (cid:105) = 0 ∀ α ∈ Σ( P, A ) } M ( ξ ) = Z G ( A ( ξ )) P ( ξ ) = P M ( ξ ) ω ( ξ ) = I M ( ξ ) M ( ξ ) ∩ P (cid:0) ω ⊗ χ | χ | − (cid:1) ν ( ξ ) = log | χ | (4.65)By Proposition 4.6 ω ( ξ ) is a pre-unitary tempered M ( ξ )-representation. Like in[78, § XI.9] these objects are designed to divide parabolic induction into stages: I GP ( ξ ) (cid:0) | χ | ⊗ ω ( ξ ) (cid:1) ∼ = Ind GP ( ξ ) (cid:0) δ / P ( ξ ) ⊗ | χ | ⊗ ω ( ξ ) (cid:1) ∼ = Ind GP ( ξ ) (cid:0) δ / P ( ξ ) ⊗ | χ | ⊗ Ind M ( ξ ) M ( ξ ) ∩ P (cid:0) δ / P ∩ M ( ξ ) ⊗ χ | χ | − ⊗ ω (cid:1)(cid:1) ∼ = Ind GP M ( ξ ) (cid:0) δ / P M ( ξ ) ⊗ Ind M ( ξ ) M ( ξ ) ∩ P (cid:0) δ / P ∩ M ( ξ ) ⊗ χ ⊗ ω (cid:1)(cid:1) ∼ = Ind GP M ( ξ ) (cid:0) Ind M ( ξ ) M ( ξ ) ∩ P (cid:0) δ / P M ( ξ ) ⊗ δ / P ∩ M ( ξ ) ⊗ χ ⊗ ω (cid:1)(cid:1) ∼ = Ind GP (cid:0) δ / P ⊗ χ ⊗ ω (cid:1) = I ( ξ ) (4.66)We say that ( P, A, ω, χ ) ∈ Ξ + if ( P, A ) is standard and log | χ | ∈ ¯ a ∗ , + . This choiceof a ”positive cone” in Ξ is justified by the next result. Lemma 4.13
Every ξ ∈ Ξ is W -associate to an element of Ξ + . If ξ , ξ ∈ Ξ + are W -associate, then the objects A ( ξ i ) , M ( ξ i ) , P ( ξ i ) and ν ( ξ i ) are the same for i = 1 and i = 2 , while ω ( ξ ) and ω ( ξ ) are equivalent M ( ξ i ) -representations.Proof. Every p-pair is conjugate to a standard p-pair, and by [61, Section 1.15]every W -orbit in a ∗ contains a unique point in positive chamber a ∗ , + . This provesthe first claim, and it also shows thatlog | χ | = log | χ | ∈ a ∗ (4.67)Hence the ν ’s, A ’s and M ’s are the same for i = 1 and i = 2. Because∆( P i , A i ) = { α (cid:12)(cid:12) a ∗ : α ∈ ∆ , (cid:104) log | χ i | , α (cid:105) > } (4.68)24 Chapter 4. Reductive p -adic groupswe must also have P ( ξ ) = P ( ξ ). If now w ∈ W is such that wω ∼ = ω , then byTheorem 4.8, applied to M ( ξ i ), there is a unitary intertwiner between ω ( ξ ) and ω ( ξ ) . (cid:50) Some immediate consequences of the above definitions and Theorem 4.7 are:
Proposition 4.14
Take ξ = ( P, A, ω, χ ) ∈ Ξ + .1. Let τ be an irreducible direct summand of ω ( ξ ) . Then ( P ( ξ ) , τ, ν ( ξ )) ∈ Λ + .2. The functor I GP ( ξ ) induces an isomorphism End G ( I ( ξ )) ∼ = End M ( ξ ) ( ω ( ξ ))
3. The irreducible quotients of I ( ξ ) are precisely the modules J ( P ( ξ ) , τ, ν ( ξ )) with τ as above. Theorem 4.15
For every π ∈ Irr( G ) there exists a unique association class W ( P, A, ω, χ ) ∈ Ξ / W such that the following equivalent statements hold :1. π is equivalent to an irreducible quotient of I ( ξ + ) , for some ξ + ∈ W ( P, A, ω, χ ) ∩ Ξ + .2. π is equivalent to an irreducible subquotient of I ( P, A, ω, χ ) , and P is maxi-mal for this property.Proof.
1. Let (
Q, σ, ν ) be the Langlands datum associated to π . Write W M , Ξ M etcetera for W , Ξ, but now corresponding to M instead of G . By Theorem 4.9there exists a unique association class W M ξ = W M ( P, A, ω, χ ) ∈ Ξ Mu / W M such that σ is a direct summand of I M ( ξ ) = I MP ( ω ⊗ χ ). Pick ξ + ∈ W M ξ ∩ Ξ + .By Proposition 4.14.3 π is equivalent to an irreducible quotient of I ( ξ + ), and byLemma 4.13 and Theorem 4.9 the class W ξ + = W ξ ∈ Ξ / W is unique for thisproperty.2. Suppose that ξ (cid:48) = ( P (cid:48) , A (cid:48) , ω (cid:48) , χ (cid:48) ) ∈ Ξ + and that π is equivalent to asubquotient of I ( ξ (cid:48) ) which is not a quotient. By Theorem 4.7.3 we have ν ( ξ (cid:48) ) − ν ( ξ ) ∈ a ( ξ (cid:48) ) ∗ , + and A ( ξ ) (cid:40) A ( ξ (cid:48) ). For α ∈ ∆ we have (cid:104) ν ( ξ (cid:48) ) , α (cid:105) = 0 ⇒ (cid:104) ν ( ξ ) , α (cid:105) = 0so by (4.65) A (cid:40) A (cid:48) and P (cid:41) P (cid:48) . Therefore the conditions 1 and 2 are equivalent. (cid:50) .4. Noncommutative geometry 125 Now that we know quite something about the representation theory of reductive p -adic groups, we can turn to the study of their noncommutative geometry withmore confidence. More specifically, we compare the periodic cyclic homologies of H ( G ) and S ( G ), and the K -theory of C ∗ r ( G ). Although these results are newand technical, this section remains very short, because we already did most ofthe hard work. We will discuss these comparison theorems in relation with theBaum-Connes conjecture.First we will prove an analogue of (3.143) for the Hecke algebra of a reductive p -adic group. Since S ( G ) is defined as an inductive limit of Fr´echet algebras, wetake its periodic cyclic homology with respect to the completed inductive tensorproduct ⊗ . Theorem 4.16
Let s ∈ B ( G ) be a Bernstein component and K s a compact opensubgroup of G as in Proposition 4.2.3. The Chern character for S ( G, K s ) s inducesan isomorphism K ∗ (cid:0) C ∗ r ( G ) s (cid:1) ⊗ C ∼ −−→ HP ∗ (cid:0) S ( G ) s , ⊗ (cid:1) The direct sum of these maps, over all s ∈ B ( G ) , is a natural isomorphism K ∗ ( C ∗ r ( G )) ⊗ C ∼ −−→ HP ∗ ( S ( G ) , ⊗ ) Proof.
Since S ( G, K s ) s is a direct summand of S ( G, K s ), by Theorem 4.11 thereare finitely many components ( P i , A i , ω i ) of Ξ such that S ( G, K s ) s ∼ = (cid:76) i (cid:0) C ∞ ( X unr ( M i )) ⊗ L ( ω i , P i ) K s × K s (cid:1) W i C ∗ r ( G, K s ) s ∼ = (cid:76) i (cid:0) C ( X unr ( M i )) ⊗ L ( ω i , P i ) K s × K s (cid:1) W i (4.69)Note that the single Bernstein component s generally contains more than compo-nent of Ξ. According to Theorem 2.13 the inclusion induces an isomorphism K ∗ (cid:0) S ( G, K s ) s (cid:1) ∼ −−→ K ∗ (cid:0) C ∗ r ( G, K s ) s (cid:1) (4.70)and by Theorem 2.27 the Chern character induces an isomorphism K ∗ (cid:0) S ( G, K ) s (cid:1) ⊗ C ∼ −−→ HP ∗ (cid:0) S ( G, K ) s (cid:1) (4.71)For any compact open K ⊂ K s the algebra L ( ω i , P i ) K × K is finite dimensional andsimple, so the inclusion L ( ω i , P i ) K s × K s → L ( ω i , P i ) K × K is of the type M n ( C ) ⊂ M m ( C ). Therefore S ( G, K s ) s → S ( G, K ) s (4.72)26 Chapter 4. Reductive p -adic groupsinduces an isomorphism on HH ∗ , HC ∗ , HP ∗ and K ∗ . From this, (4.26) and theproperties of the topological K -functor (cf. page 39) we see that K ∗ ( C ∗ r ( G )) ∼ = K ∗ (cid:32) lim −→ S (cid:77) s ∈ S C ∗ r ( G ) s (cid:33) ∼ = (cid:77) s ∈ B ( G ) K ∗ ( C ∗ r ( G ) s ) ∼ = (cid:77) s ∈ B ( G ) lim −→ K ∗ ( C ∗ r ( G, K ) s ) ∼ = (cid:77) s ∈ B ( G ) K ∗ (cid:0) C ∗ r ( G, K s ) s (cid:1) ∼ = (cid:77) s ∈ B ( G ) K ∗ (cid:0) S ( G, K s ) s (cid:1) (4.73)Since the algebras S ( G, K ) s are all nuclear Fr´echet, and have the same Hochschildhomology for fixed s ∈ B ( G ) and K ⊂ K s , we may use the continuity and ad-ditivity of HP ∗ ( · , ⊗ ), as described on page 32. (The cohomological dimension of H ( G ) s and S ( G ) s is bounded independently of s .) HP ∗ ( S ( G ) , ⊗ ) ∼ = (cid:77) s ∈ B ( G ) HP ∗ ( S ( G ) s , ⊗ ) ∼ = (cid:77) s ∈ B ( G ) lim −→ HP ∗ ( S ( G, K ) s , ⊗ ) ∼ = (cid:77) s ∈ B ( G ) HP ∗ ( S ( G, K s ) s , ⊗ )= (cid:77) s ∈ B ( G ) HP ∗ ( S ( G, K s ) s , (cid:98) ⊗ ) (4.74)Now the theorem follows from the combination of (4.71), (4.73) and (4.74). (cid:50) The analogue of Theorem 3.32 was suggested in [7, Conjecture 8.9] and in [3,Conjecture 1] :
Theorem 4.17
The inclusions H ( G ) s → S ( G ) s induce isomorphisms HP ∗ ( H ( G ) s ) ∼ −−→ HP ∗ ( S ( G ) s , ⊗ ) HP ∗ ( H ( G )) ∼ −−→ HP ∗ ( S ( G ) , ⊗ ).4. Noncommutative geometry 127 Proof.
Just as in (4.74) we have HP ∗ ( H ( G )) ∼ = (cid:77) s ∈ B ( G ) HP ∗ ( H ( G ) s ) ∼ = (cid:77) s ∈ B ( G ) lim −→ HP ∗ ( H ( G, K ) s ) ∼ = (cid:77) s ∈ B ( G ) HP ∗ ( H ( G, K s ) s ) (4.75)Therefore we only have to show that every inclusion H ( G, K s ) s → S ( G, K s ) s induces an isomorphism on periodic cyclic homology. Number the direct sum-mands in (4.69), such that S ( G, K s ) s ∼ = n s (cid:77) j =1 (cid:0) C ∞ ( X unr ( M j )) ⊗ L ( ω j , P j ) K s × K s (cid:1) W j (4.76)and dim M i ≤ dim M j if i ≤ j . Now we construct two chains of ideals H ( G, K s ) s = I ⊃ I ⊃ · · · ⊃ I n s = 0 S ( G, K s ) s = J ⊃ J ⊃ · · · ⊃ J n s = 0 I j = { h ∈ H ( G, K s ) s : I ( P j , A j , ω j , χ )( h ) = 0 if χ ∈ X nr ( M j ) and j ≤ i } J j = { h ∈ S ( G, K s ) s : I ( P j , A j , ω j , χ )( h ) = 0 if χ ∈ X unr ( M j ) and j ≤ i } Writing V i = I GP i ( E i ) K s , we clearly have J i − /J i ∼ = (cid:0) C ∞ ( X unr ( M i )) ⊗ End V i (cid:1) W i From now on we can follow the proof of Theorem 3.32. We have to substituteTheorems 4.7, 4.8, 4.11 and 4.15 for, respectively, Theorems 3.7, 3.23, 3.25 and3.31. (cid:50) .This theorem should be compared with the work of Meyer [91].It is also interesting to compare Theorems 4.16 and 4.17 with other homologicalinvariants of reductive p -adic groups. One such is the chamber homology of theBruhat-Tits building βG , equivariant with respect to G . This is a sequence ofcomplex vector spaces H Gn ( βG ) , n = 0 , , , . . . , first defined in [5, § HP i ( H ( G )) ∼ = (cid:77) n ∈ Z H Gi +2 n ( βG ) (4.77)28 Chapter 4. Reductive p -adic groupsClosely related is the G -equivariant K -homology of βG , as defined in [5, § ch G ∗ : K G ∗ ( βG ) → H G ∗ ( βG ) (4.78)which becomes an isomorphism after tensoring with C . Notice that Voigt uses analternative but equivalent definition of H G ∗ ( βG ). The Baum-Connes conjecturefor reductive p -adic groups asserts that the so-called assembly map µ : K Gj ( βG ) → K j ( C ∗ r ( G )) (4.79)is an isomorphism. This was proved by Lafforgue [79], as a part of a much moregeneral result. Putting all these things together we more or less arrive at [7,Proposition 9.4] : Theorem 4.18
In following diagram the horizontal maps are natural isomor-phisms, and the vertical maps become natural isomorphisms after tensoring with C . K G ∗ ( βG ) −→ K ∗ ( C ∗ r ( G )) ↓ ↓ H G ∗ ( βG ) ∼ = HP ∗ ( H ( G )) → HP ∗ ( S ( G ) , ⊗ )Because of the naturality, the diagram is probably commutative, but the authordoes not know how to prove this. Unfortunately the definitions are so complicatedthat it already is difficult to find any element of K G ∗ ( βG ) for which the diagramcan be seen to commute by direct computation.For the groups GL n ( F ) and SL n ( F ) partial results in the direction of Theorem4.18 were proved by Baum, Higson and Plymen in [6]. In fact, in [6] the Baum-Connes conjecture for these groups is proved precisely with the above diagram.However, the argument uses the commutativity of the diagram in an essential way,and unfortunately the authors do not provide any support of the (implicit) claimthat it does commute. hapter 5 Parameter deformations inaffine Hecke algebras
So far we have always written affine Hecke algebras as deformations of a groupalgebra, but we have not really done anything with this. Ideally speaking, severalproperties of an affine Hecke algebra H ( R , q ) should be independent of the pa-rameters q ( s ). This intuitive idea comes from finite dimensional algebras, whereit is very clear from Tits’ deformation theorem. Roughly speaking, it tells us thatif two semisimple algebras can be continuously deformed into each other, thenthey are isomorphic. This is so because there are only countably many isomor-phism classes of such algebras, and they lie discrete in some sense. For infinitedimensional algebras nothing similar holds, so there we have to find more subtleinvariants and arguments.This has been done for affine Hecke algebras with equal labels. Kazhdan andLusztig [76] gave a complete geometric parametrization of the irreducible represen-tations of such algebras. This parametrization is independent of q ∈ C × , except ina few tricky cases where q is a proper root of unity. Baum and Nistor [8] showedthat this leads to an isomorphism HP ∗ (cid:0) H ( R , q ) (cid:1) ∼ = HP ∗ (cid:0) C [ W ] (cid:1) (5.1)Acknowledging that these results cannot readily be carried over to the unequallabel case, we follow another path, more analytic in nature. By careful estimatesin the Schwartz algebra S ( R , q ) we show that the following things all depend con-tinuously on q : the operator norm, multiplication, inverting and the holomorphicfunctional calculus.Equipped with these tools and the knowledge from Chapter 3 we attack aspecial kind of parameter deformation, scaling the label function. Such deforma-tions were studied first by Opdam [98]. On the level of central characters thisamounts to scaling the absolute value by a real factor (cid:15) . For (cid:15) > C ∗ -algebras φ (cid:15) : S ( R , q (cid:15) ) ∼ −−→ S ( R , q ) (5.2)They depend continuously on (cid:15) , but the limit φ : S ( W ) → S ( R , q ) (5.3)is no longer surjective. Nevertheless this map seems to behave well. In view of(5.1) one is naturally led to conjecture that HP ∗ ( φ ) : HP ∗ (cid:0) S ( W ) (cid:1) → HP ∗ (cid:0) S ( R , q ) (cid:1) (5.4)is an isomorphism for any positive label function q . We provide various equivalentreformulations of this statement.Conjecture (5.4) can be derived from the stronger conjecture, originally due toBaum, Connes and Higson [5], that K ∗ (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = K ∗ (cid:0) C ∗ r ( W ) (cid:1) (5.5)We show that (5.5) is equivalent to the existence of a natural bijection betweenthe Grothendieck groups of irreducible representations of C ∗ r ( W ) and of C ∗ r ( R , q ).At the end of the chapter we give some clues in support of these conjectures. We recall what is already known about deformations of Iwahori-Hecke algebrasobtained by varying the label function q . For Hecke algebras of finite type thisis very clear: as long as they are semisimple they are rigid under deformations.But this is specific for the finite case, as it relies on the classification of finitedimensional semisimple algebras.For any extended Iwahori-Hecke algebra H ( R , q ) with equal labels a completeparametrization of irreducible representations is available. This is a refinementof the Langlands classification, and it is essentially independent of q . The linkbetween different q ’s is made via Lusztig’s asymptotic Hecke algebra J , whichallows a weakly spectrum preserving morphism H ( R , q ) → J . From this we willsee that the periodic cyclic homology of H ( R , q ) is independent of q , as long as itis not a proper root of unity.Recall that an algebra A is semisimple if its Jacobson radical is 0, which meansthat for every nonzero a ∈ A there is an irreducible A -representation π such that π ( a ) (cid:54) = 0. For example, by [41, Th´eor`eme 2.7.3] every C ∗ -algebra is semisimple.The structure of finite dimensional semisimple algebras is described in a famoustheorem of Wedderburn [135] :.1. The finite dimensional and equal label cases 131 Theorem 5.1
Let A be a finite dimensional semisimple algebra over a field F .There exist natural numbers n i and division algebras D i over F such that A ∼ = r (cid:77) i =1 M n i ( D i ) If F is algebraically closed then D i = F ∀ i . Let G be any finite group. By Maschke’s theorem the group algebra C [ G ] issemisimple. Let { T g : g ∈ G } be its canonical basis, and k = C [ x , . . . , x r ] apolynomial ring over C . Let A be a k -algebra whose underlying k -module is k [ G ]and whose multiplication is defined by T g · T h = (cid:88) w ∈ G a g,h,w T w (5.6)for certain a g,h,w ∈ k . For any point q ∈ C r we can endow the vector space C [ G ]with the structure of an associative algebra by T g · q T h = (cid:88) w ∈ G a g,h,w ( q ) T w (5.7)We denote the resulting algebra by H ( G, q ). It is isomorphic to the tensor product A ⊗ k C where C has the k -module structure obtained from evaluating at q . Assumemoreover that there exists a q ∈ C r such that H ( G, q ) = C [ G ]We express the rigidity of finite dimensional semisimple algebras by the followingspecial case of Tits’s deformation theorem [27, p. 357 - 359]: Theorem 5.2
There exists a polynomial P ∈ k such that the following are equiv-alent : • P ( q ) (cid:54) = 0 • H ( G, q ) is semisimple • H ( G, q ) ∼ = C [ G ]Now let ( W, S ) be a finite Coxeter system, q a label function om W and H ( W, q )the associated Iwahori-Hecke algebra, as in Section 3.1. This is consistent withthe above notation. We want to know under which conditions this algebra issemisimple. Clearly this is the case if q ( w ) > ∀ w ∈ W , for then H ( W, q ) is a C ∗ -algebra by (3.81).But the polynomials P ( q ) of Theorem 5.2 have also been determined explicitly.If we are in the equal label case q ( s ) = q ∀ s ∈ S then we may take P ( q ) = q (cid:88) w ∈ W q (cid:96) ( w ) (5.8)32 Chapter 5. Parameter deformations in affine Hecke algebrasexcept that we must omit the factor q if W is of type ( A ) n , see [54]. Moregenerally, Gyoja [53, p. 569] showed that if ( W, S ) is irreducible and S consists oftwo conjugacy classes, then in most cases we may take P ( q , q ) = q | W | q W ( q , q ) W ( q − , q ) W ( q , q ) = (cid:80) w ∈ W q ( w ) (5.9)So generically there is an isomorphism H ( W, q ) ∼ = C [ W ] (5.10)We will see later how it can be constructed explicitly. From our somewhat simplepoint of view this is all there is to say about parameter deformations of finitedimensional Hecke algebras. If they are semisimple then they are isomorphic, andif not, then they have nilpotent ideals and look very different from C [ W ].Let R = ( X, Y, R , R ∨ , F ) be a root datum, let q ∈ C × , and consider theaffine Hecke algebra with equal labels H ( R , q ). The irreducible representationsof this algebra have been classified completely by Kazhdan and Lusztig [76]. Forthis very deep result they showed among others that H ( R , q ) is isomorphic to theequivariant algebraic K -theory of a certain variety.Let G be the unique complex reductive algebraic group with root datum R ∨ = ( Y, X, R ∨ , R ), and g its Lie algebra. For reasons of a much more gen-eral nature G is called the Langlands dual group. Then T = Hom Z ( X, C × ) canbe identified with a maximal torus of G . Since every semisimple element of G isconjugate to an element of T , and since N G ( T ) /Z G ( T ) ∼ = W , we can parametrizethe central character of an (irreducible) H ( R , q )-module by a unique conjugacyclass of semisimple elements in G . So, let s ∈ G be semisimple and write n ( s, q ) = { N ∈ g : N nilpotent , Ad( s ) N = qN } (5.11)The G -conjugacy classes of pairs ( s, N ) with N ∈ n ( s, q ) are called Deligne-Langlands parameters. They give an almost complete description of Prim( H ( R , q )).For instance it works perfectly if R is of type GL n , see [137]. In a sense this isequivalent to the Langlands classification in Theorem 4.7. To find really all irre-ducible representations we must add one extra ingredient. Let Z ( s, N ) = { g ∈ G : gs = sg, Ad( g ) N = N } (5.12)be the simultaneous centralizer of s and N , and Z ( s, N ) its identity component.Assume that q ∈ C × is not a proper root of unity, i.e. either q = 1 or q is not aroot of unity. In these cases there is a bijection between Prim( H ( R , q )) and G -conjugacy classes of triples ( s, N, ρ ), where s ∈ G is semisimple, N ∈ n ( s, q ) and ρ is a ”geometric” irreducible representation of the finite group Z ( s, N ) /Z ( s, N ).This was proved for q = 1 in [72, Theorem 4.1] and for q not a root of unity and.1. The finite dimensional and equal label cases 133 X equal to the weight lattice of R ∨ in [76, Theorem 7.12]. Later it was shown in[105, Theorem 2] that this condition on X is not necessary.Another construction which is particular for the equal label case is Lusztig’sasymptotic Hecke algebra [84, 85]. This is a finite type algebra J with a basis { t w : w ∈ W } over C . It decomposes as a finite direct sum of two-sided ideals J = (cid:76) | R +0 | i =0 J i J i = span { t w : a ( w ) = i } (5.13)where a is Lusztig’s a -function. For every q ∈ C × there is an injective morphismof finite type algebras φ q : H ( R , q ) → J (5.14)If q is not a proper root of unity, then φ q induces a bijection on irreducible represen-tations. Namely, for any irreducible J -representation π the H ( R , q )-representation π ◦ φ q has a unique irreducible constituent of minimal ” a -weight”. This impliesthat the morphisms of finite type algebras φ − q (cid:16) (cid:77) i ≥ k J i (cid:17)(cid:46) φ − q (cid:16) (cid:77) i>k J i (cid:17) −→ J k (5.15)are spectrum preserving. Lusztig [85, Corollary 3.6] proved this in the case W = W aff , but using the aforementioned result of Reeder [105] his proof can be extendedto general root data. Combining this with Theorem 2.7 and Lemma 2.3 we arriveat an extended version of [8, Theorem 11] : Theorem 5.3
Assume that q is not a proper root of unity. Then HP ∗ ( φ q ) : HP ∗ ( H ( R , q )) → HP ∗ ( J ) is an isomorphism. Consequently HP ∗ ( H ( R , q )) ∼ = HP ∗ ( C [ W ])It is expected that an asymptotic Hecke algebra can also be constructed forfinite or affine Coxeter systems with unequal labels [88, Chapter 18]. Assumingcertain conjectures [88, Chapter 15] one can construct algebra homomorphisms φ q : H ( W, q ) → J for any label function with the following property: there exist v ∈ C × and n s ∈ N such that q ( s ) = v n s ∀ s ∈ S .For finite W the map φ q is an isomorphism if and only if H ( W, q ) is semisimple[88, (20.1.e)]. In this way one can find explicit formulas for the isomorphisms fromTheorem 5.2.For affine
W φ q has a nilpotent kernel [88, Proposition 18.12] and in general itis not surjective. It is unknown whether φ q is spectrum preserving in any sense.The problem is that in general there is no definite classification of all irreduciblerepresentations of an affine Hecke algebra. Apparently the link with the Langlandsdual group is much weaker for unequal parameters.34 Chapter 5. Parameter deformations in affine Hecke algebrasNevertheless in some cases the Deligne-Langlands philosophy outlined abovecan be generalized. Namely, along these lines a classification of the irreduciblerepresentations of H ( R , q ) has been obtained in [74] for R of type B n /C n , foralmost all label functions q . In this section we lay the analytic foundations for all our coming results on pa-rameter deformations. In Section 3.2 we defined various norms on an affine Heckealgebra H ( R , q ): the norm (cid:107) · (cid:107) τ associated with the trace τ , the operator norm (cid:107) · (cid:107) o and the Schwartz norms p n . We will show that the operator norm, the mul-tiplication and the inverse of an element depend continuously on q . From thiswe deduce that the holomorphic functional calculus on affine Hecke algebras iscontinuous in a very general sense.We also reconstruct the Schwartz algebra S ( R , q ) in a different way. With thisconstruction we can show in a straightforward fashion that S ( R , q ) is holomorphi-cally closed in C ∗ r ( R , q ).With respect to the bases { N w : w ∈ W } the norms (cid:107)·(cid:107) τ and p n are inde-pendent of q . Therefore we can identify all the Hilbert spaces H ( R , q ) and all theSchwartz spaces S ( R , q ) by means of this basis. When we want to consider them inthis way, only as topological vector spaces and without a specified label function,we write H ( R ) and S ( R ). To indicate that x ∈ S ( R ) should be considered as anelement of S ( R , q ) we sometimes denote it by ( x, q ). Furthermore, to distinguishthe various products we add a subscript, so · q is the multiplication in C ∗ r ( R , q ).Let L R be the space of label functions on R satisfying the positivity Condition3.8. Recall that for a simple reflection s i ∈ S aff we put q i = q ( s i ) and η i = q / i − q − / i (5.16)These numbers determine q uniquely, and their domain is only limited by theconditions q i > q i = q j whenever s i and s j are conjugate in W . Hencethe parameter space L R is homeomorphic to R n for a certain n . We will use thestandard topology on L R , induced by the metric ρ ( q, q (cid:48) ) = max s i ∈ S aff | η i − η (cid:48) i | (5.17)We already know that the group W with the length function N is of polynomialgrowth, but we need a more explicit estimate on the number of elements of a fixedlength. Lemma 5.4
There exists a real number C N such that ∀ n ∈ N (cid:8) w ∈ W : N ( w ) = n (cid:9) < C N ( n + 1) rk( X ) − Proof.
Let r denote the rank of X , and pick a linear bijection L : X ⊗ R → R r such that.2. Estimating norms 135 • L ( X ) ⊂ Z r • ∀ x ∈ X + : N ( x ) = (cid:107) L ( x ) (cid:107) This is possible because, on X + , N is additive and takes values in N . By (3.25)we can write any w ∈ W as w = uxv with u, v ∈ W , x ∈ X + If N ( w ) = n , then clearly n − | R | ≤ n − N ( u ) − N ( v ) ≤ N ( x ) ≤ n + N ( u ) + N ( v ) ≤ n + | R | (5.18)Therefore we can estimate | W | − (cid:8) w ∈ W : N ( w ) = n (cid:9) ≤ (cid:8) x ∈ X + : n − | R | ≤ N ( x ) ≤ n + | R | (cid:9) = (cid:8) y ∈ L ( X + ) : n − | R | ≤ (cid:107) y (cid:107) ≤ n + | R | (cid:9) ≤ (cid:8) y ∈ Z r : n − | R | ≤ (cid:107) y (cid:107) ≤ n + | R | (cid:9) For the sake of calculation we assume now that n > | R | . This is allowed becausethere are only finitely many w ∈ W of smaller length. We continue our estimate: ≤ ( n + | R | + 1) r − ( n − | R | − r = r (cid:88) i =0 (cid:16) ri (cid:17) n r − i (cid:0) | R | + 1 (cid:1) i (cid:0) − ( − i (cid:1) ≤ n r − (cid:88) i ≤ r, i odd (cid:16) ri (cid:17) (cid:0) | R | + 1 (cid:1) i < n r − (cid:0) | R | + 2 (cid:1) r So our candidate for C N is | W | (cid:0) | R | + 2 (cid:1) r We only have to check whether it works also for n ≤ | R | and, if not, increase itaccordingly. (cid:50) Put b = rk( X ) + 1. By Lemma 5.4 the following sum converges to a limit C b : C b := (cid:88) w ∈ W ( N ( w ) + 1) − b < ∞ (cid:88) n =0 C N ( n + 1) rk( X ) − ( n + 1) − rk( X ) − < ∞ (5.19)This implies that for any x = (cid:80) u x u N u ∈ S ( R ) and n ∈ N (cid:88) u | x u | ( N ( u ) + 1) n ≤ (cid:88) u sup v (cid:8) | x v | ( N ( v ) + 1) n + b (cid:9) ( N ( u ) + 1) − b = C b p n + b ( x )(5.20) (cid:107) x (cid:107) τ ≤ (cid:88) u | x u | ≤ C b p b ( x ) (5.21)36 Chapter 5. Parameter deformations in affine Hecke algebrasSince (cid:96) ( w ) ≤ N ( w ) ∀ w ∈ W , these inequalities a fortiori remain valid if we replace N by (cid:96) .Let u, v, w ∈ W and let u = ωs · · · s (cid:96) ( u ) be a reduced expression, where (cid:96) ( ω ) =0 and s i ∈ S aff . (The s i need not all be different.) For I ⊂ { , , . . . , (cid:96) ( u ) } we put η I = (cid:81) i ∈ I η i and u I = ω ˜ s · · · ˜ s (cid:96) ( u ) where ˜ s i = (cid:26) s i if i / ∈ Ie if i ∈ I Theorem 5.5 N u · q N v = (cid:88) I ⊂{ , ,...,(cid:96) ( u ) } η I D uv ( I ) N u I v where • D uv ( I ) is either 0 or 1 • D vu ( ∅ ) = 1 and D uv ( I ) = 0 if | I | > | R +0 |• (cid:80) I ⊂{ , ,...,(cid:96) ( u ) } D uv ( I ) < (cid:96) ( u ) + 1) | R +0 | Proof.
It follows from the multiplicaton rules (3.37) that N s i · q N v = N s i v + D s i v ( i ) η i N v where D s i v ( i ) = (cid:26) (cid:96) ( s i v ) > (cid:96) ( v )1 if (cid:96) ( s i v ) < (cid:96) ( v ) (5.22)The expression for N u · q N v , with D vu ( I ) being 0 or 1 and D vu ( ∅ ) = 1, follows fromthis, with induction to (cid:96) ( u ). By [83, Theorem 7.2] for fixed w ∈ W the sum (cid:88) I : u I = w η I D uv ( I )is a polynomial of degree at most | R +0 | in the η i . Therefore D uv ( I ) = 0 whenever | I | > | R +0 | . Consequently (cid:88) I ⊂{ , ,...,(cid:96) ( u ) } D uv ( I ) ≤ (cid:8) I ⊂ { , , . . . , (cid:96) ( u ) } : | I | ≤ | R +0 | (cid:9) ≤ | R +0 | (cid:88) j =0 (cid:18) (cid:96) ( u ) j (cid:19) ≤ (cid:96) ( u )! (cid:0) (cid:96) ( u ) − | R +0 | (cid:1) ! | R +0 | (cid:88) j =0 j ! < (cid:96) ( u ) + 1) | R +0 | (5.23)where we should interpret (cid:0) (cid:96) ( u ) − | R +0 | (cid:1) ! as 1 if | R +0 | ≥ (cid:96) ( u ) . (cid:50) Let η > C η = 3 C b max (cid:8) , η | R +0 | (cid:9) ..2. Estimating norms 137 Proposition 5.6
For all q, q (cid:48) ∈ B ρ ( q , η ) , x ∈ S ( R ) the following estimates hold. (cid:107) λ ( x, q ) (cid:107) B ( H ( R )) = (cid:107) ( x, q ) (cid:107) o ≤ C η p b + | R +0 | ( x ) (cid:107) λ ( x, q ) − λ ( x, q (cid:48) ) (cid:107) B ( H ( R )) ≤ ρ ( q, q (cid:48) ) C η p b + | R +0 | ( x ) In particular S ( R , q ) ⊂ C ∗ r ( R , q ) and every finite dimensional C ∗ r ( R , q ) -representation is tempered.Proof. Let y = (cid:80) v y v N v ∈ S ( R ). By (5.20) and Theorem 5.5 we have (cid:107) x · q y (cid:107) τ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) u,v x u y v N u · q N v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) u,v x u y v (cid:88) I η I D uv ( I ) N u I v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ ≤ (cid:88) u | x u | (cid:88) I : | I |≤| R +0 | | η I | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) v | y v | N u I v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ ≤ (cid:88) u | x u | ( (cid:96) ( u ) + 1) | R +0 | (cid:8) , η | R +0 | (cid:9) (cid:107) y (cid:107) τ ≤ C η p b + | R +0 | ( x ) (cid:107) y (cid:107) τ (5.24)Since S ( R ) is dense in H ( R ) this gives the estimate, by the very definition ofthe operator norm on B ( H ( R )). In particular we get a continuous embedding S ( R , q ) → C ∗ r ( R , q ), so every finite dimensional representation of the latter algebrais tempered by Lemma 3.14. (cid:107) x · q y − x · q (cid:48) y (cid:107) τ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) u,v x u y v ( N u · q N v − N u · q (cid:48) N v ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) u,v x u y v (cid:88) I ( η I − η (cid:48) I ) D uv ( I ) N u I v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) u,v x u y v (cid:88) I ρ ( q, q (cid:48) ) | I | η | I |− D uv ( I ) N u I v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ ≤ ρ ( q, q (cid:48) ) (cid:88) u | x u | (cid:88) I : | I |≤| R +0 | | I | η | I |− (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) v | y v | N u I v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ ≤ ρ ( q, q (cid:48) ) (cid:88) u | x u | ( (cid:96) ( u ) + 1) | R +0 | (cid:8) , η | R +0 | (cid:9) (cid:107) y (cid:107) τ ≤ ρ ( q, q (cid:48) ) C η p b + | R +0 | ( x ) (cid:107) y (cid:107) τ (5.25)38 Chapter 5. Parameter deformations in affine Hecke algebrasBetween lines 4 and 5 we used a small calculation like (5.23) : (cid:88) I : | I |≤| R +0 | | I | η | I |− ≤ | R +0 | (cid:88) j =0 (cid:18) (cid:96) ( u ) j (cid:19) jη j − ≤ (cid:96) ( u )! (cid:0) (cid:96) ( u ) − | R +0 | (cid:1) ! | R +0 | (cid:88) j =0 jj ! max (cid:8) , η | R +0 |− (cid:9) ≤ ( (cid:96) ( u ) + 1) | R +0 | (cid:8) , η | R +0 | (cid:9) (5.26)Note that we did not really use that y lies in the subspace S ( R ) of H ( R ), it onlyhelps to ensure that all intermediate expressions are well-defined. (cid:50) Now we can also estimate the behaviour of the Schwartz norms p n under mul-tiplication. Put b (cid:48) = 2 b + | R +0 | = 2 rk( X ) + | R +0 | + 2. Proposition 5.7
Let n ∈ N , q, q (cid:48) ∈ B ρ ( q , η ) and x i = (cid:80) u ∈ W x iu N u ∈ S ( R , q ) .Then p n ( x · q · · · · q x m ) ≤ m (cid:81) i =1 C η C b p n + b (cid:48) ( x i ) p n ( x · q · · · · q x m − x · q (cid:48) · · · · q (cid:48) x m ) ≤ ρ ( q, q (cid:48) ) m (cid:81) i =1 C η C b p n + b (cid:48) ( x i ) Proof.
This can be deduced with a piece of careful bookkeeping: p n ( x · q · · · · q x m ) ≤ p n ( (cid:80) u i ∈ W x u · · · x mu m N u · q · · · · q N u m ) ≤ (cid:80) u i ∈ W | x u · · · x mu m | ( N ( u ) + · · · + N ( u m ) + 1) n (cid:81) mi =1 (cid:107) ( N u i , q ) (cid:107) o ≤ (cid:80) u i ∈ W | x u · · · x mu m | (cid:81) mi =1 C η ( N ( u i ) + 1) n + b + | R +0 | = (cid:81) mi =1 C η (cid:80) u ∈ W | x iu | ( N ( u ) + 1) n + b + | R +0 | ≤ (cid:81) mi =1 C η C b p n + b (cid:48) ( x i ) p n ( N u · q · · · · q N u m − N u · q (cid:48) · · · · q (cid:48) N u m ) ≤ (cid:80) m − j =1 p n ( N u · q · · · · q N u j · q N u j +1 · q (cid:48) · · · · q (cid:48) N u m − N u · q · · · · q N u j · q (cid:48) N u j +1 · q (cid:48) · · · · q (cid:48) N u m ) ≤ (cid:80) m − j =1 ρ ( q, q (cid:48) ) (cid:81) mi =1 C η ( N ( u i ) + 1) n + b + | R +0 | ≤ ρ ( q, q (cid:48) ) (cid:81) mi =1 C η ( N ( u i ) + 1) n + b + | R +0 | p n ( x · q · · · · q x m − x · q (cid:48) · · · · q (cid:48) x m ) ≤ (cid:80) u i ∈ W | x u · · · x mu m | p n ( N u · q · · · · q N u m − N u · q (cid:48) · · · · q (cid:48) N u m ) ≤ (cid:80) u i ∈ W | x u · · · x mu m | ρ ( q, q (cid:48) ) (cid:81) mi =1 C η ( N ( u i ) + 1) n + b + | R +0 | = ρ ( q, q (cid:48) ) (cid:81) mi =1 C η (cid:80) u ∈ W | x iu | ( N ( u ) + 1) n + b + | R +0 | ≤ ρ ( q, q (cid:48) ) (cid:81) mi =1 C η p n + b (cid:48) ( x i ).2. Estimating norms 139In these calculations we used (5.20) and Proposition 5.6 several times. (cid:50) Knowing how to handle multiple products in S ( R , q ), we can even make somerough estimates for the holomorphic functional calculus. Corollary 5.8
Let f : z → (cid:80) ∞ m =0 a m z m be a holomorphic function on a neigh-borhood of ∈ C and define another holomorphic function ˜ f (with the same radiusof convergence) by ˜ f ( z ) = (cid:80) ∞ m =0 | a m | z m . For any n ∈ N , x ∈ S ( R , q ) and q, q (cid:48) ∈ B ρ ( q , η ) such that f ( x, q ) and f ( x, q (cid:48) ) are defined we have p n ( f ( x, q )) ≤ ˜ f (cid:0) C η C b p n + b (cid:48) ( x ) (cid:1) p n ( f ( x, q ) − f ( x, q (cid:48) )) ≤ ρ ( q, q (cid:48) ) ˜ f (cid:0) C η C b p n + b (cid:48) ( x ) (cid:1) Proof.
By Proposition 5.7 we have p n ( f ( x, q )) = p n (cid:32) ∞ (cid:88) m =0 a m ( x, q ) m (cid:33) ≤ ∞ (cid:88) m =0 | a m | p n (( x, q ) m ) ≤ ∞ (cid:88) m =0 | a m | (cid:0) C η C b p n + b (cid:48) ( x ) (cid:1) m = ˜ f (cid:0) C η C b p n + b (cid:48) ( x ) (cid:1) p n ( f ( x, q ) − f ( x, q (cid:48) )) = p n (cid:32) ∞ (cid:88) m =0 a m (( x, q ) m − ( x, q (cid:48) ) m ) (cid:33) ≤ ∞ (cid:88) m =0 | a m | p n (( x, q ) m − ( x, q (cid:48) ) m ) ≤ ∞ (cid:88) m =0 | a m | ρ ( q, q (cid:48) ) (cid:0) C η C b p n + b (cid:48) ( x ) (cid:1) m = ρ ( q, q (cid:48) ) ˜ f (cid:0) C η C b p n + b (cid:48) ( x ) (cid:1) The right hand sides of these inequalities might be infinite, but that is no problem. (cid:50)
With this result we can show that inverting is continuous as a function of x and q . Proposition 5.9
The set of invertible elements (cid:83) q ∈ L R S ( R , q ) × is open in S ( R ) × L R , and inverting is a continuous map from this set to itself.
40 Chapter 5. Parameter deformations in affine Hecke algebras
Proof.
First we recall that if (cid:107) ( z, q ) (cid:107) o <
1, then z is invertible in C ∗ r ( R , q ), withinverse (cid:80) ∞ n =0 (1 − z ) n . Take q, q (cid:48) ∈ B ρ ( q , η ) , y ∈ S ( R ) , x ∈ S ( R , q ) × and write a = ( x, q ) − . If the sum converges, then a · q (cid:48) ∞ (cid:88) m =1 (1 − ( x + y ) · q (cid:48) a, q (cid:48) ) m = a · q (cid:48) (( x + y ) · q (cid:48) a, q (cid:48) ) − − a · q (cid:48) x + y, q (cid:48) ) − − a (5.27)By Proposition 5.7 p n (( x + y ) · q (cid:48) a − ≤ p n ( x · q (cid:48) a − x · q a ) + p n ( y · q (cid:48) a ) ≤ ρ ( q, q (cid:48) ) C η C b p n + b (cid:48) ( x ) p n + b (cid:48) ( a ) + C η C b p n + b (cid:48) ( y ) p n + b (cid:48) ( a ) (5.28)Let U be the open neighborhood of ( x, q ) consisting of those( x + y, q (cid:48) ) ∈ S ( R ) × B ρ ( q , η ) for which ρ ( q, q (cid:48) ) C η C b p b + | R | ( x ) p b + | R | ( a ) < / C η C b p b + | R | ( y ) p b + | R | ( a ) < / (cid:107) (( x + y ) · q (cid:48) a − , q (cid:48) ) (cid:107) o < ∀ ( x + y, q (cid:48) ) ∈ U so every element of U is invertible. To show the continuity of inverting we considerthe function f ( z ) = ∞ (cid:88) m =1 z m = z/ (1 − z )By (5.27) and Corollary 5.8 we have p n (( x + y, q (cid:48) ) − − a ) ≤ C b C η p n + b (cid:48) ( a ) p n + b (cid:48) (cid:0) f (1 − ( x + y ) · q (cid:48) a, q (cid:48) ) (cid:1) ≤ C b C η p n + b (cid:48) ( a ) f (cid:0) C b C η p n +2 b (cid:48) (1 − ( x + y ) · q (cid:48) a ) (cid:1) Since f (0) = 0 we deduce from (5.28) that this expression is small whenever ρ ( q, q (cid:48) )and y are small. (cid:50) With this result we can see that the holomorphic functional calculus is contin-uous in the most general sense.
Corollary 5.10
Let V ⊂ S ( R ) , Q ⊂ L R and U ⊂ C be open subsets such thatthe spectrum of every ( x, q ) ∈ V × Q is contained in U . Then the map C an ( U ) × V × Q → S ( R ) : ( f, x, q ) → f ( x, q ) is continuous. .2. Estimating norms 141 Proof.
Recall from Theorem 2.9.4 that f ( x, q ) = 12 πi (cid:90) Γ f ( λ )( λ − x, q ) − dλ for a suitable contour Γ ⊂ U around sp( x, q ). By Corollary 3.27.2 sp( x, q ) is com-pact, and by Proposition 5.9 it depends continuously on x and q . Therefore wecan find a contour which is suitable for every ( x (cid:48) , q (cid:48) ) in a small neighborhood of( x, q ). Now apply Proposition 5.9 and Theorem 2.9.3. (cid:50) Let H ( R ) ∗ be the algebraic dual of H ( R ), which we identify, using the bitrace τ , with the space of all formal infinite sums (cid:80) w ∈ W x w N w . The length function N may also be considered as an endomorphism of H ( R ) ∗ : λ ( N ) : (cid:88) w ∈ W x w N w → (cid:88) w ∈ W N ( w ) x w N w (5.29)This is an unbounded operator on H ( R ), but it does restrict to a continuousendomorphism of S ( R ). For T ∈ B ( H ( R )) put D ( T )= [ λ ( N ) , T ]. Inspired by thework of Vign´eras [130, Section 7] we study the space V ∞N ( R , q ) = { x ∈ H ( R ) ∗ : D n ( λ ( x )) ∈ B ( H ( R )) ∀ n ∈ Z ≥ } (5.30)We use the topology defined by the collection of seminorms (cid:8) (cid:107) D n ( λ ( · )) (cid:107) B ( H ( R )) , n ∈ Z ≥ (cid:9) (5.31)In fact we already know this space: Lemma 5.11 V ∞N ( R , q ) = S ( R , q ) Proof.
From the proof of Proposition 5.6 we see that for any y = (cid:80) v ∈ W y v N v ∈ H ( R ) , n ∈ Z ≥ , u ∈ W (cid:107) D n ( λ ( N u )) y (cid:107) τ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) v y v n (cid:88) i =0 ( − i (cid:16) ni (cid:17) λ ( N ) n − i λ ( N u ) λ ( N ) i N v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) v y v n (cid:88) i =0 ( − i (cid:16) ni (cid:17) (cid:88) I η I D uv ( I ) N ( u I v ) n − i N ( v ) i N u I v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) v y v (cid:88) I η I D uv ( I )( N ( u I v ) − N ( v )) n N u I v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ ≤ N ( u ) n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) v | y v | (cid:88) I | η I | D uv ( I ) N u I v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ ≤ N ( u ) n N ( u ) + 1) | R +0 | max (cid:8) , η | R +0 | (cid:9) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) v | y v | N u I v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ = N ( u ) n N ( u ) + 1) | R +0 | max (cid:8) , η | R +0 | (cid:9) (cid:107) y (cid:107) τ
42 Chapter 5. Parameter deformations in affine Hecke algebraswhere η = ρ ( q, q ). Hence, for x = (cid:80) u x u N u ∈ H ( R ) ∗ (cid:107) D n ( λ ( x )) (cid:107) o = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) u x u D n ( λ ( N u )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) o ≤ (cid:88) u | x u | N ( u ) + 1) n + | R + o | max (cid:8) , η | R +0 | (cid:9) ≤ C η p n + b + | R +0 | ( x )On the other hand, since Ω (cid:48) = { ω ∈ W : N ( ω ) = 0 } is finite, p n ( x ) ≤ (cid:88) u ∈ W ( N ( u ) + 1) n | x u | ≤ (cid:88) ω ∈ Ω (cid:48) | x ω | + 4 n (cid:88) u ∈ W N ( u ) n | x u | ≤ | Ω (cid:48) | (cid:107) x (cid:107) τ + 4 n (cid:107) λ ( N ) n x (cid:107) τ = | Ω (cid:48) | (cid:107) λ ( x ) N e (cid:107) τ + 4 n (cid:107) D n ( λ ( x )) N e (cid:107) τ ≤ | Ω (cid:48) | (cid:107) λ ( x ) (cid:107) o + 4 n (cid:107) D n ( λ ( x )) (cid:107) o ≤ (cid:0) | Ω (cid:48) | / (cid:107) λ ( x ) (cid:107) o + 2 n (cid:107) D n ( λ ( x )) (cid:107) o (cid:1) Therefore the collections of seminorms { p n : n ∈ Z ≥ } and (5.31) are equivalent. (cid:50) So we found a different way to construct the Schwartz algebra of an affineHecke algebra. An advantage of this construction is that it allows us to proveCorollary 3.27 in a more elementary way, relying only on the density of V ∞N ( R , q )in C ∗ r ( R , q ) and not on any representation theory. Theorem 5.12 V ∞N ( R , q ) is a complete locally convex algebra with jointlycontinuous multiplication.2. V ∞N ( R , q ) × is open in V ∞N ( R , q ) , and inverting is a continuous map fromthis set to itself.3. An element of V ∞N ( R , q ) is invertible if and only if it is invertible in C ∗ r ( R , q ) .Proof.
1. By Lemma 5.11 V ∞N ( R , q ) is a Fr´echet space. Since D is a derivation, itis also a topological algebra with jointly continuous multiplication.2. See [130, Lemma 16]. Suppose that x ∈ V ∞N ( R , q ) and (cid:107) x (cid:107) o <
1. Then1 − x ∈ C ∗ r ( R , q ) × and (1 − x ) − = (cid:80) ∞ n =0 x n ∈ C ∗ r ( R , q ). We have to show thatthis sum converges in V ∞N ( R , q ). For n, r ∈ N D r ( λ ( x )) = (cid:88) r + ··· + r n = r r ! D r ( λ ( x )) · · · D r n ( λ ( x )) r ! · · · r n !.2. Estimating norms 143Every product D r ( λ ( x )) · · · D r n ( λ ( x )) contains at least n − r factors λ ( x ) and atmost r factors of the form D i ( λ ( x )) with i >
0. Therefore (cid:107) D r ( λ ( x )) (cid:107) B ( H ( R )) ≤ n r M r (cid:107) x (cid:107) no M = max (cid:110)(cid:13)(cid:13) D i ( λ ( x )) (cid:13)(cid:13) B ( H ( R )) (cid:107) x (cid:107) − o : 0 < i ≤ r (cid:111) This gives (cid:13)(cid:13) D r ( λ (1 − x ) − ) (cid:13)(cid:13) B ( H ( R )) ≤ ∞ (cid:88) n =0 n r M r (cid:107) x (cid:107) no ≤ r ! M r (1 − (cid:107) x (cid:107) o ) − r − from which we conclude that indeed (1 − x ) − ∈ V ∞N ( R , q ) and that inverting in V ∞N ( R , q ) is continuous around 1. In general, if y ∈ V ∞N ( R , q ) × then we can usethe ”translation” λ ( y − ) to show that V ∞N ( R , q ) × contains an open neighborhoodof y and that inverting is continuous on this set.3. Suppose that z ∈ V ∞N ( R , q ) ∩ C ∗ r ( R , q ) × . By Lemma 5.11 V ∞N ( R , q ) is dense in C ∗ r ( R , q ), so we can find y ∈ z − B ∩ Bz − , where B = { x ∈ C ∗ r ( R , q ) : (cid:107) − x (cid:107) o < } By the above yz, zy ∈ B ∩ V ∞N ( R , q ) ⊂ V ∞N ( R , q ) × so z is also invertible in V ∞N ( R , q ) . (cid:50) Note that it does not follow from these considerations that V ∞N ( R , q ) is a m-algebra. To prove that we still have to use Theorem 3.25.Consider the bundle of Banach spaces (cid:70) q ∈ L R C ∗ r ( R , q ) over L R . For any fixed x ∈ S ( R ) the constant function q → x is a section of this bundle, and by Propo-sition 5.6 the function q → (cid:107) ( x, q ) (cid:107) o is continuous on L R . So by [41, Proposition10.2.3] there is a unique collection Γ of sections of (cid:70) q ∈ L R C ∗ r ( R , q ) containing allthese constant sections, which makes this into a field of C ∗ -algebras, in the senseof Dixmier [41, Section 10.3]. By construction Γ contains all continuous maps L R → S ( R ). In particular for any compact set Q ⊂ L R we can construct the uni-tal C ∗ -algebra C ∗ r ( R , Q )= Γ (cid:12)(cid:12) Q , which contains C ( Q ; S ( R )) as a dense subalgebra.For a related construction, assume that Q is a smooth submanifold of L R , notnecessarily compact. Although the author is not aware of any precise definition,he believes it makes sense to call (cid:70) q ∈ L R S ( R , q ) a field of Fr´echet algebras. ByProposition 5.7 the set of smooth sections S ( R , Q ) = C ∞ ( Q ; S ( R )) (5.32)is a Fr´echet space and a topological algebra with jointly continuous multiplication.However, the author does not know whether S ( R , Q ) is a m-algebra in general.44 Chapter 5. Parameter deformations in affine Hecke algebrasFor every q ∈ Q we can obtain submultiplicative seminorms on S ( R , q ) from theFourier transform, but these are not easily expressible in terms of the elements N w .Therefore it is not clear whether we can choose them in some sense continuous in q and ”glue” such seminorms to a family of seminorms that defines the topologyof S ( R , Q ).Seminorms that we can construct are related to the principal series representa-tions, which exist for every q ∈ L R . From Theorem 3.3.1 we get a linear bijection H ( R , q ) → H ( W , q ) ⊗ C [ X ]This can be extended to a continuous map φ θ,q : S ( R , q ) → H ( W , q ) ⊗ S ( X ) (5.33)which is essentially the direct integral of all unitary principal series representations.In general φ θ,q is neither injective nor surjective. For n ∈ N we define the followingnorm on H ( W , q ) ⊗ S ( X ) : σ n (cid:16) (cid:88) w ∈ W ,x ∈ X y w,x N w θ x (cid:17) = (cid:88) w ∈ W ,x ∈ X ( N ( x ) + 1) n (5.34)We compose it with φ θ,q to get the seminorm σ n,q = σ n ◦ φ θ,q on S ( R , q ). Theseseminorms are continuous in q : Lemma 5.13
Let n ∈ N , η > and q, q (cid:48) ∈ B ρ ( q , η ) . There exists a real number C n,η such that for all z ∈ S ( R ) σ n,q ( z ) ≤ C n,η p n + b + | R +0 | ( z ) | σ n,q ( z ) − σ n,q (cid:48) ( z ) | ≤ ρ ( q, q (cid:48) ) C n,η p n + b + | R +0 | ( z ) Proof.
Let x + ∈ X + and put P = { α ∈ F : α ∨ ( x + ) = 0 } By [61, Proposition 1.15] we have W P = { w ∈ W : wx + = x + } and hence W x + W = W P x + W = W x + (cid:0) W P (cid:1) − From (3.41) we see that the N u with u ∈ W P commute with θ x + ∈ H ( R , q ). Pickany w ∈ W x + W . From Theorem 5.5 we see that there is a unique way to write N w = (cid:88) u ∈ W ,v ∈ W P c wu,v N u θ x + N v − (5.35)where every c wu,v is a polynomial in the η i of degree at most | R | . From the lengthformula in Proposition 3.1 we see that the c wu,v depend only on P , in the following.2. Estimating norms 145sense. If w = w x + w ( w , w ∈ W ) and w (cid:48) = w x (cid:48) w where x (cid:48) ∈ X + and { α ∈ F : α ∨ ( x (cid:48) ) = 0 } = P , then c w (cid:48) u,v = c wu,v . In particular there are only finitelymany different c wu,v , less than | W | | F | .Assume for simplicity that η ≥
1, so (cid:107) ( N v , q ) (cid:107) o ≤ (2 η ) | R +0 | ∀ v ∈ W Let K be an upper bound for the absolute values of all c wu,v , also under the condition q ∈ B ρ ( q , η ). From a repeated application of (3.41) we see that θ x + N v − equalsa sum of at most (2 N ( x + ) + 2) | R +0 | terms of the form η I N v (cid:48) θ x , where N ( x ) ≤N ( x + ) , v (cid:48) ∈ W and I is a multi-index with | I | ≤ | R +0 | . This leads to the estimate σ n,q ( N w ) = σ n (cid:16) (cid:88) u,v c wu,v N u (cid:88) I,v (cid:48) ,x η I N v (cid:48) θ x (cid:17) ≤ (cid:88) v | W | / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) u c wu,v N u (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ (cid:88) I,v (cid:48) ,x (cid:107) ( N v (cid:48) , q ) (cid:107) o | η I | ( N ( x ) + 1) n ≤ | W | K (2 η ) | R +0 | η | R +0 | (2 N ( x + ) + 2) | R +0 | ( N ( x + ) + 1) n = | W | K (2 η ) | R | ( N ( x + ) + 1) n + | R +0 | ≤ | W | K (2 η ) | R | ( | R +0 | + 1) n + | R +0 | ( N ( w ) + 1) n + | R +0 | For z = (cid:80) w ∈ W z w N w ∈ S ( R ) we obtain σ n,q ( z ) ≤ (cid:88) w ∈ W | z w | σ n,q ( N w ) ≤ | W | K (2 η ) | R | ( | R +0 | + 1) n + | R +0 | (cid:88) w ∈ W | z w | ( N ( w ) + 1) n + | R +0 | ≤ | W | K (2 η ) | R | ( | R +0 | + 1) n + | R +0 | C b p n + b + | R +0 | ( z )Plugging the description of θ x + N v − into (5.35) we see that N w = (cid:88) u ∈ W ,x ∈ X y wu,x N u θ x (5.36)where the y wu,x are polynomials in the η i of degree at most 2 | R | . Therefore wecan write φ θ,q ( z ) = (cid:88) I : | I |≤ | R | (cid:88) u ∈ W ,x ∈ X η I z I,u,x N u θ x := (cid:88) I η I z I (5.37)Since the finite collection { η I : | I | ≤ | R |} is linearly independent, considered asfunctions of the η i , we can find constants K n such that σ n ( z I ) ≤ K n p n + b + | R +0 | ( z ) ∀ I
46 Chapter 5. Parameter deformations in affine Hecke algebrasHence | σ n,q ( z ) − σ n,q (cid:48) ( z ) | ≤ (cid:88) I : | I |≤ | R | | η I − η (cid:48) I | σ n ( z I ) ≤ ρ ( q, q (cid:48) )2 | R | η | R | K n p n + b + | R +0 | ( z )To finish the proof we take for C n,η the maximum of | W | K (2 η ) | R | ( | R +0 | + 1) n + | R +0 | C b and 2 | R | η | R | K n . (cid:50) We conclude this section with an important remark. All the estimates obtainedhere can be generalized to M k ( S ( R , q )) for any k ∈ N . Of course we first have to(re)define p ( k ) n (cid:0) z i,j ) ki,j =1 (cid:1) = max ≤ i,j ≤ k p n ( z i,j ) (5.38)but then all our results can be extended by standard techniques. In the comingsections we assume that this has been done, and we attach a superscript ( k ) tothe modified constants. Fix a root datum R and a label function q . Instead of considering general deforma-tions of q , we concentrate on the scaled label functions q (cid:15) for (cid:15) ∈ [ − , H ( R , q (cid:15) ) can be ”scaled” accordingly. We will construct isomorphisms S ( R , q (cid:15) ) → S ( R, q ) for (cid:15) > S ( W ) = S ( R , q ) → S ( R , q ), alldepending continuously on (cid:15) . This requires a lot of long calculations, which relyon the technical parts of Chapter 3 and Section 5.2.Recall from [98, Section 6.3] that we always have a canonical isomorphism H ( R , q ) ∼ −−→ H ( R , q − ) : N w → ( − (cid:96) ( w ) N w (5.39)This map preserves * and τ , and it extends to the Schwartz and C ∗ -completions.However, our scaling maps will not lead to this map for (cid:15) = − S ( R , q ) we want to regard it not only as an algebra ofinvariant sections (via the Fourier transform), but also as the image of a projectorin a larger algebra. Let P (cid:48) ( F ) be a complete set of representatives for the actionof W on the power set of F . By Theorem 3.25 there are canonical direct sumdecompositions S ( R , q ) = (cid:76) P ∈P (cid:48) ( F ) S ( R , q ) P = (cid:76) P ∈P (cid:48) ( F ) S ( R , q ) e P C ∗ r ( R , q ) = (cid:76) P ∈P (cid:48) ( F ) C ∗ r ( R , q ) P = (cid:76) P ∈P (cid:48) ( F ) C ∗ r ( R , q ) e P (5.40).3. Scaling the labels 147where the e P are central idempotents in S ( R , q ). But this can be refined. Let∆ (cid:48) be a collection of representatives for the action of W on ∆. For ( P, δ ) ∈ ∆,we let W δ be the isotropy group of ( P, δ ) in W . The Fourier transform gives anisomorphism F (cid:48) : S ( R , q ) ∼ −−→ (cid:77) ( P,δ ) ∈ ∆ (cid:48) C ∞ (cid:0) T Pu ; End (cid:0) H ( W P ) ⊗ V δ (cid:1)(cid:1) W δ := (cid:77) ( P,δ ) ∈ ∆ (cid:48) A W δ δ (5.41)We must be careful when taking invariants, since W δ → A × δ : g → π ( g, P, δ, · ) (5.42)is not necessarily a group homomorphism. In fact, by (3.129) it is a projectiverepresentation. By Schur’s theorem [113] there exists a finite central extension { e } → N δ → Γ δ → W δ → { e } (5.43)such that every projective representation of W δ lifts to a unique linear represen-tation of Γ δ . This lift does not depend on the ˜ δ g that we chose in (3.112) toconstruct π ( g, ξ ) for ξ = ( P, δ, t ). In fact, the problems with (5.42) arise only fromthe ambiguity in the definition of ˜ δ g . Lifting things to a linear representation ofΓ δ is therefore equivalent to picking, for every lift γ ∈ Γ δ of g ∈ W δ , a multiple ˜ δ γ of ˜ δ g such that Γ δ → A × δ : γ → π ( γ, P, δ, · )becomes multiplicative. Writing u γ ( ξ ) = π ( γ, γ − ξ )(3.129) becomes the cocycle relation u γγ (cid:48) = u γ u γγ (cid:48) (5.44)Notice the similarity with the proof of Lemma 2.26. Consider the crossed product A δ (cid:111) Γ δ ∼ = End (cid:0) H ( W P ) ⊗ V δ (cid:1) ⊗ C ∞ (cid:0) T Pu (cid:1) (cid:111) Γ δ (5.45)with respect to the action of Γ δ on T Pu . Lemma A.1 gives some information aboutthis algebra. We still need to determine A W δ δ = A Γ δ δ , but using the multiplicationin A δ (cid:111) Γ δ we can write the group action as γ ( a ) = u γ γaγ − u − γ (5.46)Moreover by (5.44) Γ δ → (cid:0) A δ (cid:111) Γ δ (cid:1) × : γ → u γ γ is a unitary representation, so p δ ( u ) := | Γ δ | − (cid:88) γ ∈ Γ δ u γ γ ∈ A δ (cid:111) Γ δ (5.47)48 Chapter 5. Parameter deformations in affine Hecke algebrasis a projection. By Lemma A.2 the map A Γ δ δ → p δ ( u ) (cid:0) A δ (cid:111) Γ δ (cid:1) p δ ( u ) : a → p δ ( u ) ap δ ( u ) (5.48)is an isomorphism of pre- C ∗ -algebras.Consider for (cid:15) ∈ [ − ,
1] the affine Hecke algebras H (cid:15) = H ( R , q (cid:15) ) with labelfunctions q (cid:15) ( w ) = q ( w ) (cid:15) . Let c α,(cid:15) and ı ow,(cid:15) be the c -functions and normalizedintertwiners for these algebras. From the formula (3.101) we see that the residualcosets are also related by scaling in suitable directions. Their tempered forms allapproach T u when (cid:15) → r ∈ T and write r = r u exp( r s ) with r u ∈ T u and r s ∈ t rs . Let B ⊂ Lie( T )be a ball satisfying Conditions 3.21. Then (cid:15)B satisfies these conditions with respectto r u exp( (cid:15)r s ) and q (cid:15) , except that it is not open for (cid:15) = 0. (Here we use | (cid:15) | ≤ U (cid:15) = W (cid:0) r u exp( (cid:15) ( r s + B )) (cid:1) and we define a W -equivariant scaling map σ (cid:15) : U → U (cid:15) σ (cid:15) (cid:0) w ( r exp( b )) (cid:1) = w (cid:0) r u exp( (cid:15) ( r s + b )) (cid:1) (5.49)Assume now that 0 (cid:54) = (cid:15) ∈ [ − , σ (cid:15) is ananalytic diffeomorphism. We can combine it with (3.117) and Theorem 3.22 toconstruct algebra isomorphisms ρ (cid:15) : H me(cid:15) ( U (cid:15) ) ∼ −−→ H me ( U ) ρ (cid:15) (cid:32) (cid:88) w ∈ W a w ı ow,(cid:15) (cid:33) = (cid:88) w ∈ W ( a w ◦ σ (cid:15) ) ı ow a w ∈ C me ( U (cid:15) ) (5.50)We intend to show that these maps depend analytically on (cid:15) and have a well-defined limit as (cid:15) →
0. Notice that σ = lim (cid:15) → σ (cid:15) is a locally constant map withrange W r u . Lemma 5.14
For (cid:15) (cid:54) = 0 and α ∈ R write d α,(cid:15) = ( c α,(cid:15) ◦ σ (cid:15) ) c − α . This definesa bounded invertible analytic function of u and (cid:15) which extends to a function on U × [ − , with the same properties.Proof. This is an extended version of [98, Lemma 5.2]. Let us write d α,(cid:15) ( u ) = f f f f g g g g ( u ) = 1 + θ − α/ ( u )1 + θ − α/ ( σ (cid:15) ( u )) × q − (cid:15)/ α ∨ θ − α/ ( σ (cid:15) ( u ))1 + q − / α ∨ θ − α/ ( u ) 1 − θ − α/ ( u )1 − θ − α/ ( σ (cid:15) ( u )) 1 − q − (cid:15)/ α ∨ q − (cid:15) α ∨ θ − α/ ( σ (cid:15) ( u ))1 − q − / α ∨ q − α ∨ θ − α/ ( u )We see that d α,(cid:15) ( u ) extends to an invertible analytic function on U × [ − ,
1] ifnone of the quotients f n /g n has a zero or a pole on this domain. By Condition.3. Scaling the labels 1493.21.2 there is a unique b ∈ w ( r s + B ) / u = w ( r u ) exp(2 b ). This formsa coordinate system on w ( r exp( B )), and σ (cid:15) ( u ) = w ( r u ) exp(2 b(cid:15) ). By Condition3.21.4 if either f n ( u ) = 0 or g n ( u ) = 0 for some u ∈ w ( r exp( B )) ⊂ U , then f n ( w ( r )) = g n ( w ( r )) = 0. One can easily check that in this situation f n ( u ) g n ( u ) = (cid:18) − e − α ( b ) (cid:15) − e − α ( b ) (cid:19) ( − n Again by Condition 3.21.2 the only critical points of this function are those forwhich α ( b ) = 0. If (cid:15) (cid:54) = 0 then both the numerator and the denominator have azero of order 1 at such points, so the singularity is removable. For the case (cid:15) = 0we need to have a closer look. In our new coordinate system we can write c α,(cid:15) ( σ (cid:15) ( u )) = f ( u ) f ( u ) g ( u ) g ( u )= r u ( w − α/
2) + (cid:0) q − / α ∨ e − α ( b ) (cid:1) (cid:15) r u ( w − α/
2) + e − α ( b ) (cid:15) r u ( w − α/ − (cid:0) q − / α ∨ q − α ∨ e − α ( b ) (cid:1) (cid:15) r u ( w − α/ − e − α ( b ) (cid:15) Standard calculations using L’Hospital’s rule show thatlim (cid:15) → c α,(cid:15) ( σ (cid:15) ( u )) = r u ( w − α ) (cid:54) = 1 α ( b ) + log( q α ∨ ) / α ( b ) if r u ( w − α/
2) = − α ( b ) + log( q α ∨ ) + log( q α ∨ ) / α ( b ) if r u ( w − α/
2) = 1Thus at least d α, = lim (cid:15) → d α,(cid:15) exists as a meromorphic function on U . For r u ( w − α ) (cid:54) = 1 , d α, = c − α is invertible by Condition 3.21.4. For r u ( w − α/
2) = − d α, ( u ) = 1 − e − α ( b ) α ( b ) α ( b ) + log( q α ∨ ) / − q − / α ∨ e − α ( b ) e − α ( b ) q − / α ∨ q − α ∨ e − α ( b ) while for r u ( w − α/
2) = 1 d α, ( u ) = 1 − e − α ( b ) α ( b ) 1 + e − α ( b ) q − / α ∨ e − α ( b ) α ( b ) + log( q α ∨ ) + log( q α ∨ ) / − q − / α ∨ q − α ∨ e − α ( b ) These expressions define invertible functions by Condition 3.21.2. We concludethat indeed d α,(cid:15) ( u ) and d − α,(cid:15) ( u ) are analytic functions on U × [ − , (cid:50)
50 Chapter 5. Parameter deformations in affine Hecke algebrasNote that d α, = 1 and that d α, − ( u ) = r u ( w − α ) − e − α ( b ) r u ( w − α ) − e α ( b ) r u ( w − α/
2) + q / α ∨ e α ( b ) r u ( w − α/
2) + q − / α ∨ e − α ( b ) × r u ( w − α/ − q / α ∨ q α ∨ e α ( b ) r u ( w − α/ − q − / α ∨ q − α ∨ e − α ( b ) (5.51)If either r u ( w − α ) = θ − α ( w ( r u )) = 1 or q α ∨ = q α ∨ = 1 this simplifies consider-ably to d α, − ( u ) = q α ∨ q α ∨ (5.52)We can use Lemma 5.14 to show that the maps ρ (cid:15) preserve analyticity: Lemma 5.15
The isomorphisms (5.50) restrict to isomorphisms ρ (cid:15) : H an(cid:15) ( U (cid:15) ) ∼ −−→ H an ( U ) These maps have a well-defined limit homomorphism ρ = lim (cid:15) → ρ (cid:15) : C [ W ] → H an ( U ) such that for every w ∈ W the function [ − , → H an ( U ) : (cid:15) → ρ (cid:15) ( N w ) is analytic.Proof. The first statement is [98, Theorem 5.3], but for the remainder we need toprove this anyway. It is clear that ρ (cid:15) restricts to an isomorphism between C an ( U (cid:15) )and C an ( U ). For a simple reflection s ∈ S corresponding to α ∈ F we have N s + q ( s ) − (cid:15)/ = q − (cid:15)/ c α,(cid:15) ( ı os,(cid:15) + 1) ρ (cid:15) ( N s ) = q (cid:15)/ ( c α,(cid:15) ◦ σ (cid:15) )( ı os + 1) − q ( s ) − (cid:15)/ = q ( s ) ( (cid:15) − / ( c α,(cid:15) ◦ σ (cid:15) ) c − α (cid:0) N s + q ( s ) − / (cid:1) − q ( s ) − (cid:15)/ = q ( s ) ( (cid:15) − / d α,(cid:15) (cid:0) N s + q ( s ) − / (cid:1) − q ( s ) − (cid:15)/ (5.53)By Lemma 5.14 such elements are analytic in (cid:15) ∈ [ − ,
1] and u ∈ U , so in particularthey belong to H an ( U ). Moreover, since every d α,(cid:15) is invertible, the set { ρ (cid:15) ( N w ) : w ∈ W } is a basis for H an ( U ) as a C an ( U )-module. Therefore ρ (cid:15) restricts to anisomorphism between H an(cid:15) ( U (cid:15) ) and H an ( U ) for (cid:15) (cid:54) = 0.For any x ∈ X, ρ (cid:15) ( θ x ) = θ x ◦ σ (cid:15) depends analytically on (cid:15) , as a function on U . Combined with (5.35) this shows that ρ (cid:15) ( N w ) is analytic in (cid:15) ∈ [ − ,
1] for any w ∈ W . Thus ρ exists as a linear map. But, being a limit of algebra homomor-phisms, it must also be multiplicative. (cid:50) .3. Scaling the labels 151Note that ρ is never injective or surjective because σ (cid:15) is not. Moreover ρ − is not the isomorphism (5.39). In general (5.51) is not even rational, so ρ − ( N s )cannot lie in H . In the simple case r u ( w − α ) = 1 we have ρ − ( θ α ) = θ − α and, by(5.52) and (5.53) ρ − ( N s ) = N s + q ( s ) − / − q ( s ) / = N − s Usually the maps ρ (cid:15) do not preserve the *, but this can be fixed. For (cid:15) ∈ [ − , M (cid:15) = ρ (cid:15) ( N − w ,(cid:15) ) ∗ N w (cid:89) α ∈ R +1 d α,(cid:15) ∈ H an ( U )We will use M (cid:15) to extend [98, Corollary 5.7]. However, this result contained a smallmistake: the construction of the element A (cid:15) in [98] was unfortunately influencedby an inessential oversight. To correct this we replace it by M (cid:15) . Theorem 5.16
For all (cid:15) ∈ [ − , M (cid:15) is invertible, positive and bounded. It hasa positive square root in H an ( U ) and the map (cid:15) → M / (cid:15) is analytic. ˜ ρ (cid:15) : H an(cid:15) ( U (cid:15) ) → H an ( U )˜ ρ (cid:15) ( h ) = M / (cid:15) ρ (cid:15) ( h ) M − / (cid:15) is a homomorphism of topological *-algebras, and an isomorphism if (cid:15) (cid:54) = 0 . Forany w ∈ W the function [ − , → H an ( U ) : (cid:15) → ˜ ρ (cid:15) ( N w ) is analytic.Proof. By Lemmas 5.14 and 5.15 the M (cid:15) are invertible, bounded and analytic in (cid:15) . Consider, for (cid:15) (cid:54) = 0, the automorphism µ (cid:15) of H me ( U ) given by µ (cid:15) ( h ) = ρ (cid:15) ( ρ − (cid:15) ( h ) ∗ ) ∗ On one hand, for f ∈ C me ( U ) we have by definition (3.118) and the W -equivarianceof σ (cid:15) µ (cid:15) ( f ) = ρ (cid:15) (( f ◦ σ (cid:15) ) ∗ ) ∗ = ρ (cid:15) (cid:0) N w ( f − w ◦ σ (cid:15) ) N − w ,(cid:15) (cid:1) ∗ = ρ (cid:15) ( N − w ,(cid:15) ) ∗ (cid:0) f − w (cid:1) ∗ ρ (cid:15) ( N w ,(cid:15) ) ∗ = ρ (cid:15) ( N − w ,(cid:15) ) ∗ N w f N − w ρ (cid:15) ( N w ,(cid:15) ) ∗ = ρ (cid:15) ( N − w ,(cid:15) ) ∗ N w (cid:81) α ∈ R +1 d α,(cid:15) f (cid:81) α ∈ R +1 d − α,(cid:15) N − w ρ (cid:15) ( N w ,(cid:15) ) ∗ = M (cid:15) f M − (cid:15) (5.54)52 Chapter 5. Parameter deformations in affine Hecke algebrasOn the other hand, suppose that the simple reflections s and s (cid:48) = w sw ∈ S correspond to α and α (cid:48) = − w α ∈ F . Using (3.123) and (3.121) we find M (cid:15) ı os M − (cid:15) = ρ (cid:15) ( N − w ,(cid:15) ) ∗ N w (cid:81) α ∈ R +1 d α,(cid:15) ı os (cid:81) α ∈ R +1 d − α,(cid:15) N − w ρ (cid:15) ( N w ,(cid:15) ) ∗ = ρ (cid:15) ( N − w ,(cid:15) ) ∗ N w ı os d − α,(cid:15) d − α,(cid:15) N − w ρ (cid:15) ( N w ,(cid:15) ) ∗ = ρ (cid:15) ( N − w ,(cid:15) ) ∗ N w ı os c − α c α N − w N w c α,(cid:15) ◦ σ (cid:15) c − α,(cid:15) ◦ σ (cid:15) N − w ρ (cid:15) ( N w ,(cid:15) ) ∗ = ρ (cid:15) ( N − w ,(cid:15) ) ∗ ( ı s (cid:48) ) ∗ (cid:18) c α (cid:48) ,(cid:15) ◦ σ (cid:15) c − α (cid:48) ,(cid:15) ◦ σ (cid:15) (cid:19) ∗ ρ (cid:15) ( N w ,(cid:15) ) ∗ = (cid:18) ρ (cid:15) ( N w ,(cid:15) ) c α (cid:48) ,(cid:15) ◦ σ (cid:15) c − α (cid:48) ,(cid:15) ◦ σ (cid:15) ı s (cid:48) ρ (cid:15) ( N − w ,(cid:15) ) (cid:19) ∗ = ρ (cid:15) (cid:18) N w c α (cid:48) c − α (cid:48) ı os (cid:48) ,(cid:15) N − w ,(cid:15) (cid:19) ∗ = ρ (cid:15) (cid:0) ( ı os,(cid:15) ) ∗ (cid:1) ∗ = µ (cid:15) ( ı os ) (5.55)Since C me ( U ) and the ı s generate H me ( U ), we conclude that µ (cid:15) ( h ) = M (cid:15) hM − (cid:15) ∀ h ∈ H me ( U )Now we can see that ρ (cid:15) (cid:0) N − w ,(cid:15) (cid:1) ∗ = ρ (cid:15) (cid:0) ( N ∗ w ,(cid:15) ) − (cid:1) ∗ = ρ (cid:15) (cid:0) ( N − w ,(cid:15) ) ∗ (cid:1) ∗ = µ (cid:15) (cid:0) ρ (cid:15) ( N − w ,(cid:15) ) (cid:1) = M (cid:15) ρ (cid:15) ( N − w ,(cid:15) ) M − (cid:15) N e = M − (cid:15) ρ (cid:15) ( N − w ,(cid:15) ) ∗ N w (cid:81) α ∈ R +1 d α,(cid:15) = ρ (cid:15) ( N − w ,(cid:15) ) M − (cid:15) N w (cid:81) α ∈ R +1 d α,(cid:15) M (cid:15) = N w (cid:81) α ∈ R +1 d α,(cid:15) ρ (cid:15) ( N − w ,(cid:15) ) = (cid:0) ρ (cid:15) ( N − w ,(cid:15) ) ∗ (cid:0) N w (cid:81) α ∈ R +1 d α,(cid:15) (cid:1) ∗ (cid:1) ∗ = (cid:0) ρ (cid:15) ( N − w ,(cid:15) ) ∗ N w (cid:81) α ∈ R +1 d α,(cid:15) (cid:1) ∗ = M ∗ (cid:15) Thus the elements M (cid:15) are Hermitian ∀ (cid:15) (cid:54) = 0. By continuity in (cid:15) M is alsoHermitian. Moreover they are all invertible, and M = N e , so they are in factstrictly positive. We already knew that the element (cid:15) → M (cid:15) of C an ([ − , H an ( U )) ∼ = C an ([ − , × U ) ⊗ A H is bounded, so we can construct its square root using holomorphic functional cal-culus in the Fr´echet Q-algebra C anb ([ − , × U ) ⊗ A H . This ensures that (cid:15) → M / (cid:15) is still analytic. Finally, for (cid:15) (cid:54) = 0˜ ρ (cid:15) ( h ) ∗ = (cid:16) M / (cid:15) ρ (cid:15) ( h ) M − / (cid:15) (cid:17) ∗ = M − / (cid:15) ρ (cid:15) ( h ) ∗ M / (cid:15) = M − / (cid:15) µ (cid:15) ( ρ (cid:15) ( h ∗ )) M / (cid:15) = M / (cid:15) ρ (cid:15) ( h ∗ ) M − / (cid:15) = ˜ ρ (cid:15) ( h ∗ ) (5.56).3. Scaling the labels 153Again this extends to (cid:15) = 0 by continuity. (cid:50) Let Rep ( C ∗ r ( R , q )) and Rep ( S ( R , q )) be the categories of finite dimensionalrepresentations of C ∗ r ( R , q ) and of S ( R , q )). We defineRep U ( C ∗ r ( R , q )) = Rep ( C ∗ r ( R , q )) ∩ Rep U ( H ( R , q ))Rep U ( S ( R , q )) = Rep ( S ( R , q )) ∩ Rep U ( H ( R , q )) (5.57)Recall that H t(cid:15) is the residual algebra of H (cid:15) at t ∈ T , whose (finite dimensional)representations are precisely Rep W t ( C ∗ r ( R , q (cid:15) )). Lemma 5.17
For (cid:15) ∈ [ − , the map ˜ ρ (cid:15) induces a ”scaling map” ˜ σ (cid:15) : Rep W u ( H ) → Rep W σ (cid:15) ( u ) ( H (cid:15) ) which preserves unitarity and is a bijection if (cid:15) (cid:54) = 0 .For (cid:15) < σ (cid:15) exchanges tempered and anti-tempered modules. For (cid:15) ≥ σ (cid:15) preserves (anti-)temperedness and ˜ ρ (cid:15) descends to a *-homomorphism ρ (cid:15) : H σ (cid:15) ( u ) (cid:15) → H u which is an isomorphism if (cid:15) > .Proof. If π ∈ Rep( H ) and (cid:15) (cid:54) = 0 then by construction t ∈ T is an A -weight of π if and only if σ (cid:15) ( t ) is an A (cid:15) -weight of π ◦ ˜ ρ (cid:15) . The A -weights of π ◦ ˜ ρ are allcontained in W r ⊂ T u . Therefore˜ σ (cid:15) : π → π ◦ ˜ ρ (cid:15) (5.58)maps Rep W u ( H ) to Rep W σ (cid:15) ( u ) ( H (cid:15) ), for any u ∈ U and (cid:15) ∈ [ − , ρ (cid:15) is a*-homomorphism and a bijection (for (cid:15) (cid:54) = 0) ˜ σ (cid:15) has the desired properties for such (cid:15) . Moreover for x ∈ X we have | σ (cid:15) ( t )( x ) | = | t ( x ) | (cid:15) , which proves the assertionsabout temperedness. If (cid:15) ≥ π extends to C ∗ r ( R , q ) then it is unitary and tem-pered by Proposition 5.6. Therefore π ◦ ˜ ρ (cid:15) is tempered and completely reducible,and Corollary 3.26 assures that it extends to C ∗ r ( R , q (cid:15) ). This implies˜ ρ (cid:15) (cid:0) Rad σ (cid:15) ( u ) ,(cid:15) (cid:1) ⊂ Rad u so we get a *-homomorphism ρ (cid:15) : H (cid:15) / Rad σ (cid:15) ( u ) ,(cid:15) = H σ (cid:15) ( u ) (cid:15) → H / Rad u = H u If (cid:15) > ρ − (cid:15) , so ρ (cid:15) is an isomorphism. (cid:50) .54 Chapter 5. Parameter deformations in affine Hecke algebrasAssume once more that 0 (cid:54) = (cid:15) ∈ [ − , r → σ (cid:15) ( r ) is a bijection betweenthe residual points for ( R , q ) and those for ( R , q (cid:15) ). The groupoid of intertwiners W is independent of q , and it acts on the set of H (cid:15) -representations of the form π ( P, δ (cid:48) , t ) with δ (cid:48) irreducible but not necessarily discrete series. The definitions(3.114) and (3.126) also makes sense in this case. If we realize the representation π (cid:0) P, ˜ σ (cid:15) ( δ ) , t (cid:1) on H (cid:0) W P (cid:1) ⊗ V δ as Ind H (cid:15) H P(cid:15) (cid:0) δ ◦ ˜ ρ (cid:15) ◦ φ t,(cid:15) (cid:1) then we get homomorphisms F (cid:48) (cid:15) : H ( R , q (cid:15) ) → (cid:77) ( P,δ ) ∈ ∆ (cid:48) O (cid:0) T P (cid:1) ⊗ End (cid:0) H ( W P ) ⊗ V δ (cid:1) F (cid:48) (cid:15) ( h )( P, δ, t ) = π ( P, ˜ σ (cid:15) ( δ ) , t )( h ) (5.59)(Notice that this is also defined for (cid:15) = 0.) The isotropy groups W ˜ σ (cid:15) ( δ ) for various (cid:15) ’s may be identified, so the image of (5.59) consists of sections that are invariantunder a certain action of W δ . We add a subscript (cid:15) to indicate which action weconsider.Recall from (5.43) that Γ δ is a Schur extension of W δ . There are uniquelydetermined u γ,(cid:15) ∈ A × δ such that, just as in (5.46), we can write the associatedΓ δ -action as γ (cid:15) ( a ) = u γ,(cid:15) γaγ − u − γ,(cid:15) in A δ (cid:111) Γ δ (5.60) Lemma 5.18
The elements u γ,(cid:15) depend analytically on (cid:15) and u γ, = lim (cid:15) → u γ,(cid:15) exists. Moreover u γ,(cid:15) is unitary ∀ (cid:15) ∈ [ − , , γ ∈ Γ δ .Proof. Let γ be a lift of kn ∈ W δ and write ξ = γ ( P, ˜ σ (cid:15) ( δ ) , t ) ∈ Ξ u,(cid:15) . By definition(3.114) for h ∈ H ( W P ) , v ∈ V δ u γ,(cid:15) ( ξ )( h ⊗ v ) = h · q (cid:15) ı on − ,(cid:15) ( t ) ⊗ ˜ δ γ,(cid:15) ( v ) (5.61)where ˜ δ γ,(cid:15) ∈ U ( V δ ). More precisely, ˜ δ γ,(cid:15) is a multiple of a map L (cid:15) := ˜ δ kn,(cid:15) ∈ U ( V δ )that satisfies δ (˜ ρ (cid:15) h ) = L − (cid:15) δ ( ρ (cid:15) ψ k ψ n h ) L (cid:15) h ∈ H P,(cid:15) (5.62)This L (cid:15) is only defined up to scalars, but we will show that these can be chosensuch that L = lim (cid:15) → L (cid:15) exists. Since V δ is a finite dimensional vector space,every automorphism of End( V δ ) is inner, and Aut(End( V δ )) ∼ = P GL ( V δ ). Because GL ( V δ ) → P GL ( V δ ) is a fiber bundle it suffices to show that L exists as aprojective linear map. Writing h = ρ − (cid:15) h and rearranging some terms transforms(5.62) into δ ( M − / (cid:15) ) L (cid:15) δ ( M / (cid:15) ) δ ( h ) δ ( M − / (cid:15) ) L − (cid:15) δ ( M / (cid:15) ) = δ ( ρ (cid:15) ψ k ψ n ρ − (cid:15) h )If we can show that everything else in this equation is analytic in (cid:15) and has awell-defined limit as (cid:15) →
0, then the same must hold for L (cid:15) ∈ P GL ( V δ ). ByTheorem 5.16 we know this already for M ± / (cid:15) . For any f ∈ C an ( U ) we have.3. Scaling the labels 155 ρ (cid:15) ψ k ψ n ρ − (cid:15) f = ψ k ψ n f by the W -equivariance of σ (cid:15) . For the simple reflection s associated to α ∈ F we have ρ (cid:15) ψ k ψ n ρ − (cid:15) (cid:0) N s + q ( s ) − / (cid:1) = ρ (cid:15) ψ k ψ n (cid:0) q ( s ) (1 − (cid:15) ) / ( c α ◦ σ /(cid:15) c − α,(cid:15) ( N s + q ( s ) − (cid:15)/ ) (cid:1) = ρ (cid:15) ψ k (cid:0) q ( s ) (1 − (cid:15) ) / ( c nα ◦ σ /(cid:15) ) c − nα,(cid:15) ( N nsn − + q ( s ) − (cid:15)/ ) (cid:1) = ψ k (cid:0) q ( s ) (1 − (cid:15) ) / c nα ( c − nα,(cid:15) ◦ σ (cid:15) ) (cid:1) ρ (cid:15) (cid:0) N nsn − + q ( s ) − (cid:15)/ (cid:1) By Lemmas 5.14 and 5.15 the last expression has the required properties.Now we turn our attention to the other parts of (5.61). For (cid:15) > σ (cid:15) ( δ ) is discrete series. Therefore u γ,(cid:15) is unitary and ı on − ,(cid:15) cannothave a pole at t . From the explicit definition (3.120) we see that ı on − ,(cid:15) is regularat t for any (cid:15) ∈ [ − , (cid:15) → ı on − ,(cid:15) ( t ) = N n − . By Proposition 5.7 thisimplies lim (cid:15) → h · q (cid:15) ı on − ,(cid:15) ( t ) = h · q N n − Putting things together we conclude that L (cid:15) , ˜ δ γ,(cid:15) and u γ,(cid:15) are analytic in (cid:15) andhave well-defined limits as (cid:15) →
0, all as projective linear maps. However, wealready agreed that we may assume that this even holds for L (cid:15) as a linear map.But by [37, §
53] the way to lift this representation from W δ to Γ δ is completelydetermined by the cocycle W δ × W δ → C × : ( g , g ) → ˜ δ g ,(cid:15) ˜ δ g ,(cid:15) ˜ δ − g − g − ,(cid:15) This cocycle is continuous in (cid:15) and Γ δ is finite, so the way to lift is independent of (cid:15) . Therefore lim (cid:15) → ˜ δ γ,(cid:15) and lim (cid:15) → u γ,(cid:15) exist even as linear maps. It also followsthat u γ,(cid:15) is analytic in (cid:15) which, in combination with its unitarity ∀ (cid:15) >
0, showsthat it is unitary ∀ (cid:15) ∈ [ − , . (cid:50) Lemma 5.19
The pre- C ∗ -algebras A W δ,(cid:15) δ = A Γ δ,(cid:15) δ are all isomorphic, by isomor-phisms that are piecewise analytic in (cid:15), (cid:15) (cid:48) ∈ [ − , .Proof. From Lemma 5.18 we get an analytic path of projections[ − , → A δ (cid:111) Γ δ : (cid:15) → p δ ( u (cid:15) ) := | Γ δ | − (cid:88) γ ∈ Γ δ u γ,(cid:15) γ (5.63)Like in (5.48) the map A Γ δ,(cid:15) δ → p δ ( u (cid:15) )( A δ (cid:111) Γ δ ) p δ ( u (cid:15) ) : a → p δ ( u (cid:15) ) ap δ ( u (cid:15) )is an isomorphism of pre- C ∗ -algebras. If we apply [102, Lemma 1.15] we seethat the p δ ( u (cid:15) ) are all conjugate, by elements depending continuously on (cid:15) . Toshow analyticity we construct these elements explicitly, using the recipe of [10,Proposition 4.32]. For (cid:15), (cid:15) (cid:48) ∈ [ − ,
1] consider the element z ( δ, (cid:15), (cid:15) (cid:48) ) := (2 p δ ( u (cid:48) (cid:15) ) − p δ ( u (cid:15) ) −
1) + 1 ∈ A δ (cid:111) Γ δ
56 Chapter 5. Parameter deformations in affine Hecke algebrasClearly this is analytic in (cid:15) and (cid:15) (cid:48) and p δ ( u (cid:15) ) z ( δ, (cid:15), (cid:15) (cid:48) ) = 2 p δ ( u (cid:48) (cid:15) ) p δ ( u (cid:15) ) = z ( δ, (cid:15), (cid:15) (cid:48) ) p δ ( u (cid:15) )Moreover if (cid:107)·(cid:107) is the norm of the enveloping C ∗ -algebra C δ := C (cid:0) T Pu ; End (cid:0) H ( W P ) ⊗ V δ (cid:1)(cid:1) (cid:111) Γ δ of A δ (cid:111) Γ δ and (cid:107) p δ ( u (cid:15) ) − p δ ( u (cid:48) (cid:15) ) (cid:107) < (cid:107) z ( δ, (cid:15), (cid:15) (cid:48) ) − (cid:107) = (cid:107) p δ ( u (cid:48) (cid:15) ) p δ ( u (cid:15) ) − p δ ( u (cid:15) ) − p δ ( u (cid:15) ) (cid:107) = (cid:13)(cid:13)(cid:13) − (cid:0) p δ ( u (cid:15) ) − p δ ( u (cid:48) (cid:15) ) (cid:1) (cid:13)(cid:13)(cid:13) < z ( δ, (cid:15), (cid:15) (cid:48) ) is invertible in C δ . But A δ (cid:111) Γ δ is holomorphically closed in C δ , so z ( δ, (cid:15), (cid:15) (cid:48) ) is also invertible in this Fr´echet algebra. Moreover, because the Fr´echettopology on A δ (cid:111) Γ δ is finer than the topology coming from (cid:107)·(cid:107) , there exists anopen interval I (cid:15) containing (cid:15) such that z ( δ, (cid:15), (cid:15) (cid:48) ) is invertible ∀ (cid:15) (cid:48) ∈ I (cid:15) . For such (cid:15) (cid:48) we construct the unitary element u ( δ, (cid:15), (cid:15) (cid:48) ) := z ( δ, (cid:15), (cid:15) (cid:48) ) | z ( δ, (cid:15), (cid:15) (cid:48) ) | − By construction the map p δ ( u (cid:15) )( A δ (cid:111) Γ δ ) p δ ( u (cid:15) ) → p δ ( u (cid:48) (cid:15) )( A δ (cid:111) Γ δ ) p δ ( u (cid:48) (cid:15) ) : x → u ( δ, (cid:15), (cid:15) (cid:48) ) xu ( δ, (cid:15), (cid:15) (cid:48) ) − is an isomorphism of pre- C ∗ -algebras. The composite map A Γ δ,(cid:15) δ → A Γ δ,(cid:15) (cid:48) δ is givenby x → | Γ δ | (cid:2) u ( δ, (cid:15), (cid:15) (cid:48) ) p δ ( u (cid:48) (cid:15) ) xp δ ( u (cid:48) (cid:15) ) u ( δ, (cid:15), (cid:15) (cid:48) ) − (cid:3) e (5.64)which is analytic in (cid:15) and (cid:15) (cid:48) because p δ ( u (cid:15) ) is. Now we pick a finite cover { I (cid:15) i } mi =1 of [ − , (cid:15), (cid:15) (cid:48) ∈ [ − ,
1] an isomorphism between A Γ δ,(cid:15) δ and A Γ δ,(cid:15) (cid:48) δ can be obtained by composing at most m isomorphisms of the type (5.64). (cid:50) The constructions in this section lead to the following
Corollary 5.20
There exists a collection of injective *-homomorphisms φ (cid:15) : H ( R , q (cid:15) ) → S ( R , q ) (cid:15) ∈ [ − , such that1. for (cid:15) < the map Rep( S ( R , q )) → Rep( H ( R , q (cid:15) )) : π → π ◦ φ (cid:15) is a bijection from tempered H -representations to anti-tempered H (cid:15) -representations .3. Scaling the labels 157 ∀ ( P, δ, t ) ∈ Ξ u the representation π ( P, δ, t ) ◦ φ (cid:15) is equivalent with π (cid:15) ( P, ˜ σ (cid:15) ( δ ) , t ) φ is the canonical embedding4. φ (cid:15) ( N w ) = N w ∀ w ∈ Z ( W ) ∀ h ∈ H ( R ) the function [ − , → S ( R , q ) : (cid:15) → φ (cid:15) ( h ) is piecewise analytic, and in particular analytic at .Proof. By Lemma 5.18 the image of F (cid:48) is invariant under the action of W δ, . Sowe can define φ (cid:15) = F (cid:48)− ◦ ζ (cid:15) ◦ F (cid:48) (cid:15) (5.65)where ζ (cid:15) = (cid:76) ( P,δ ) ∈ ∆ (cid:48) ζ (cid:15),δ and ζ (cid:15),δ : A W δ,(cid:15) δ → A W δ δ is the isomorphism from Lemma 5.19. Now 2 and 3 are valid by construction, 4follows from the observation that the Z ( W )-character of π (cid:15) ( P, ˜ σ (cid:15) ( δ ) , t ) is equal to t (cid:12)(cid:12) Z ( W ) , for every (cid:15) ∈ [ − , (cid:15) → φ (cid:15) ( h ) is analytic at 0.As concerns the injectivity of φ (cid:15) , note that π (cid:15) ( P, ˜ σ (cid:15) ( δ ∅ ) , t ) = I t,(cid:15) is a principalseries representation for H (cid:15) . So if h ∈ ker( φ (cid:15) ), then h acts as 0 on all unitaryprincipal series. By Lemma 3.4 we must have h = 0 . (cid:50) We do not know whether φ (cid:15) ( H (cid:15) ) ⊂ H , for two reasons : ζ (cid:15),δ need not preservepolynomiality, and not every polynomial section is in the image of F (cid:48) . Theorem 5.21
For (cid:15) ∈ [0 , there exist homomorphisms of pre- C ∗ -algebras φ (cid:15) : S ( R , q (cid:15) ) → S ( R , q ) φ (cid:15) : C ∗ r ( R , q (cid:15) ) → C ∗ r ( R , q ) with the properties1. φ (cid:15) is an isomorphism if (cid:15) > and φ is injective2. ∀ ( P, δ, t ) ∈ Ξ u the representation π ( P, δ, t ) ◦ φ (cid:15) is equivalent with π (cid:15) ( P, ˜ σ (cid:15) ( δ ) , t ) φ is the identity4. φ (cid:15) ( h ) = h ∀ h ∈ S ( Z ( W )) (cid:15) → φ (cid:15) ( h ) is continuous ∀ h ∈ S ( R )58 Chapter 5. Parameter deformations in affine Hecke algebras Proof.
For any (
P, δ ) ∈ ∆ the representation ˜ σ (cid:15) ( δ ), although not necessarilyirreducible if (cid:15) = 0, is certainly completely reducible, being unitary. Hence byLemma 5.17 every irreducible constituent π of ˜ σ (cid:15) ( δ ) is a direct summand ofInd H (cid:15),P H P (cid:15),P ( P , δ , φ t ,(cid:15) )for certain P ⊂ P, δ ∈ ∆ (cid:15) and t ∈ Hom Z (cid:0) ( X P ) P , S (cid:1) = Hom Z (cid:0) X/ ( X ∩ ( P ∨ ) ⊥ + Q P ) , S (cid:1) ⊂ T u Consequently π ( P, π , t ) is a direct summand ofInd H (cid:15) H P(cid:15) (cid:16)
Ind H (cid:15),P H P (cid:15),P ( δ ◦ φ t ,(cid:15) ) ◦ φ t,(cid:15) (cid:17) = π (cid:15) ( P , δ , tt )In particular every matrix coefficient of π (cid:15) ( P, ˜ σ (cid:15) ( δ ) , t ) appears in the Fourier trans-form of H (cid:15) , and (5.59) extends to the respective Schwartz and C ∗ -completions, asrequired. By Lemma 5.17 and (5.41) these maps are isomorphisms if (cid:15) > φ remainsinjective: every irreducible tempered H ( R , q )-representation is a quotient of aunitary principal series, so any element of C ∗ r ( R , q ) that vanishes on all unitaryprincipal series is 0. Furthermore properties 2. and 4. are direct consequences ofCorollary 5.20.Finally, if x = (cid:80) w ∈ W x w N w ∈ S ( R ) then this sum converges uniformly to x .For every partial sum x (cid:48) the map (cid:15) → φ (cid:15) ( x (cid:48) ) is continuous by Corollary 5.20, sothis also holds for x itself. (cid:50) Although there were quite a few arbitrary choices involved in constructing φ (cid:15) ,the homotopy class of these maps is well-defined: Lemma 5.22
The construction of ( φ (cid:15) ) (cid:15) ∈ [0 , is unique up to a homotopy of objectswith the properties of Theorem 5.21.Proof. Let us inventorize all the choices we made in the above construction.Already in (5.41) we chose a set of representatives ∆ (cid:48) of ∆ / W . This implicitlyfixed realizations V δ of the discrete series representations δ ∈ ∆ (cid:48) , but since we neverused a basis of V δ , the construction does not really depend on this vector space.Then we used intertwiners π ( g, ξ ) , g ∈ W δ that were defined only up to scalars,but this ambiguity dropped out when we lifted things to Γ δ . Finally we chose a(monotonuous) sequence ( (cid:15) i ) mi =0 such that (cid:15) = (cid:15), (cid:15) m = 1 and every isomorphism S ( R , q (cid:15) i ) → S ( R , q (cid:15) i +1 ) involved only one map of the type (5.64) for every δ .Suppose we take another sequence ( (cid:15) (cid:48) i ) m (cid:48) i =0 . Allowing some of the (cid:15) i and (cid:15) (cid:48) j tocoincide, we may assume that m (cid:48) = m . Then we can continuously deform the firstsequence into the second. Since the elements u ( δ, (cid:15), (cid:15) (cid:48) ) depend analytically on (cid:15) and (cid:15) (cid:48) , this will give us a path of isomorphisms..3. Scaling the labels 159To see what happens when we use some ∆ (cid:48)(cid:48) instead of ∆ (cid:48) is more difficult. It isclearly enough to investigate the effect on only one component of Ξ u , correspondingto ( P, δ ) ∈ ∆ (cid:48) and to ( P (cid:48) , δ (cid:48) ) ∈ ∆ (cid:48)(cid:48) . By the previous paragraph we may also restrictourselves to (cid:15) and (cid:15) (cid:48) with | (cid:15) − (cid:15) (cid:48) | ”sufficiently” small. Then the original isomorphismis given by φ (cid:15),(cid:15) (cid:48) : a → | Γ δ | (cid:2) p δ ( u (cid:15) (cid:48) ) u ( δ, (cid:15), (cid:15) (cid:48) ) p δ ( u (cid:15) ) ap δ ( u (cid:15) ) u ( δ, (cid:15), (cid:15) (cid:48) ) − p δ ( u (cid:15) (cid:48) ) (cid:3) e (5.66)and the alternative by φ (cid:48) (cid:15),(cid:15) (cid:48) : a (cid:48) → | Γ δ (cid:48) | (cid:2) u ( δ (cid:48) , (cid:15), (cid:15) (cid:48) ) p δ (cid:48) ( u (cid:15) ) a (cid:48) p δ (cid:48) ( u (cid:15) ) u ( δ (cid:48) , (cid:15), (cid:15) (cid:48) ) − (cid:3) e (5.67)In both formulas [ · ] e means taking the coefficient at the identity element in somegroup algebra. To compare φ (cid:15),(cid:15) (cid:48) and φ (cid:48) (cid:15),(cid:15) (cid:48) we take g ∈ W δδ (cid:48) , ξ = ( P, δ, t ) ∈ Ξ u andevaluate φ (cid:48) (cid:15),(cid:15) (cid:48) a ( ξ ), assuming that a and φ (cid:48) (cid:15),(cid:15) (cid:48) ( a ) are g -invariant: φ (cid:48) (cid:15),(cid:15) (cid:48) a ( ξ ) = g − ( φ (cid:48) (cid:15),(cid:15) (cid:48) ( ga ))( ξ ) = π (cid:48) (cid:15) ( g − , gξ ) (cid:0) φ (cid:48) (cid:15),(cid:15) (cid:48) ( ga ) (cid:1) ( gξ ) π (cid:48) (cid:15) ( g − , gξ ) − = | Γ δ (cid:48) | π (cid:48) (cid:15) ( g − , gξ ) (cid:2) u ( δ (cid:48) , (cid:15), (cid:15) (cid:48) ) p δ (cid:48) ( u η ) g ( a ) p δ (cid:48) ( u (cid:15) ) u ( δ (cid:48) , (cid:15), (cid:15) (cid:48) ) − (cid:3) e ( gξ ) π (cid:48) (cid:15) ( g − , gξ ) − = | Γ δ (cid:48) | π (cid:48) (cid:15) ( g − , gξ ) (cid:2) u ( δ (cid:48) , (cid:15), (cid:15) (cid:48) )( gξ ) p δ (cid:48) ( u (cid:15) )( gξ ) π (cid:15) ( g, ξ ) a ( ξ ) ◦ π (cid:15) ( g, ξ ) − p δ (cid:48) ( u (cid:15) )( gξ ) u ( δ (cid:48) , (cid:15), (cid:15) (cid:48) ) − ( gξ ) (cid:3) e π (cid:48) (cid:15) ( g − , gξ ) − (5.68)To compare Γ δ and Γ δ (cid:48) we have to make the Schur extension (5.43) functorial.In general it is not even unique, but the recipe in [37, §
53] always works. Forthe sake of the argument we might temporarily redefine the Schur extension to bethe result of this construction. As a bonus it is easily seen that for γ (cid:48) ∈ Γ δ (cid:48) theelement γ = g − γ (cid:48) g ∈ Γ δ is well-defined. But then necessarily π ( g, ξ ) − u γ (cid:48) ( gξ ) π ( g, ξ ) = u γ ( ξ )since both sides are proportional and satisfy the cocycle relations (5.44). In par-ticular p δ (cid:48) ( u )( gξ ) π ( g, ξ ) = π ( g, ξ ) p δ ( u )( ξ ) π (cid:48) (cid:15) ( g − , gξ ) u ( δ (cid:48) , (cid:15), (cid:15) (cid:48) )( gξ ) π η ( g, ξ ) = (2 p δ (cid:48) ( u (cid:15) (cid:48) )( ξ ) − b ( ξ )(2 p δ (cid:48) ( u (cid:15) )( ξ ) −
1) + b ( ξ )We denote the last expression by u ( b )( ξ ), where b ( ξ ) = π (cid:48) (cid:15) ( g − , gξ ) π (cid:15) ( g, ξ ). Ingeneral elements of the type u ( b ) in a C ∗ -algebra are invertible if (cid:107) p δ (cid:48) ( u (cid:15) ) − p δ (cid:48) ( u (cid:15) (cid:48) ) (cid:107) + 4 (cid:107) b − (cid:107) < p δ (cid:48) ( u (cid:15) (cid:48) ) u ( b ) = p δ (cid:48) ( u (cid:15) (cid:48) ) u ( b ) p δ (cid:48) ( u (cid:15) ) = u ( b ) p δ (cid:48) ( u (cid:15) )60 Chapter 5. Parameter deformations in affine Hecke algebrasWith this knowledge we can rewrite (5.68) as φ (cid:48) (cid:15),(cid:15) (cid:48) a ( ξ ) = | Γ δ | [ p δ ( u (cid:15) (cid:48) ) u ( b ) p δ ( u (cid:15) ) ap δ ( u (cid:15) ) u ( b ) − p δ ( u (cid:15) (cid:48) )] e ( ξ )but now e is an element of Γ δ instead of Γ δ (cid:48) . Comparing this to (5.66) we findthat the only difference is that u ( δ, (cid:15), (cid:15) (cid:48) ) has been replaced by this u ( b ). The in-tertwiners π (cid:15) (cid:48) ( g, ξ ) depend analytically on (cid:15) (cid:48) , so we can always find a path from u ( b ) to u ( δ, (cid:15), (cid:15) (cid:48) ) consisting only of elements of the same type. (This step mightrequire a subdivision of the interval between (cid:15) and (cid:15) (cid:48) , but that is no problem.)The corresponding isomorphisms form a path from φ (cid:48) (cid:15),(cid:15) (cid:48) to φ (cid:15),(cid:15) (cid:48) . (cid:50) K -theoretic conjectures We saw in Section 5.2 that the multiplication in S ( R , q ) varies continuously with q . Since a class in K -theory is rigid under small perturbations, it is natural toexpect that the K -groups of S ( R , q ) are independent of q . We reformulate thisconjecture in terms of the map φ and show that it implies some other importantconjectures.Our main tools are the scaling maps φ (cid:15) from Theorem 5.21. From Theorems5.21 and 2.13, Lemma 5.22 and the homotopy invariance of K -theory we see thatfor all (cid:15) ∈ [0 ,
1] the map K ∗ ( φ (cid:15) ) : K ∗ ( C ∗ r ( R , q (cid:15) )) ∼ = K ∗ ( S ( R , q (cid:15) )) → K ∗ ( C ∗ r ( R , q )) ∼ = K ∗ ( S ( R , q )) (5.69)is natural. By (3.143) the same goes for HP ∗ ( φ (cid:15) ) : HP ∗ ( S ( R , q (cid:15) )) → HP ∗ ( S ( R , q )) (5.70)Obviously, by Theorem 5.21 these maps are isomorphisms for (cid:15) >
0, but whetherthis holds in general for (cid:15) = 0 is not known.Let G ( C ∗ r ( R , q )) be the Grothendieck group of the additive category of finitedimensional C ∗ r ( R , q )-modules. There is a natural map G ( φ ) : G (cid:0) C ∗ r ( R , q ) (cid:1) → G (cid:0) C ∗ r ( W ) (cid:1) G ( φ )( π, V ) = ( π ◦ φ , V ) (5.71)For U ⊂ T /W we introduce the two-sided ideals J sU = { x ∈ S ( R , q ) : π ( P, W P r, δ, t )( x ) = 0 if t ∈ T Pu , W rt ∈ U } J cU = { x ∈ C ∗ r ( R , q ) : π ( P, W P r, δ, t )( x ) = 0 if t ∈ T Pu , W rt ∈ U } (5.72)By Theorem 5.21.2 φ factors through φ W t : S ( W ) /J sW t → S ( R , q ) /J sW t T rs (5.73).4. K -theoretic conjectures 161The induced map on K +0 can be regarded as a morphism of semigroups K +0 (cid:0) φ W t (cid:1) : Rep W t (cid:0) C ∗ r ( W ) (cid:1) → Rep W t T rs (cid:0) C ∗ r ( R , q ) (cid:1) (5.74)The direct sum of these maps, over all W t ∈ T u /W , is a homomorphism K Rep ( φ ) : G (cid:0) C ∗ r ( W ) (cid:1) → G (cid:0) C ∗ r ( R , q ) (cid:1) (5.75)This map is a bit weird, it does not always preserve dimensions, and it certainlyis not an inverse of G ( φ ). For example, consider the case R = A , X the rootlattice and q ( s ) = q ( s ) >
1. Then (cid:0) K Rep ( φ ) π (cid:1) ◦ φ = π if π admits a central character t (cid:54) = 1. The only irreducible C W -representationswith central character t = 1 are the trivial and sign representations of W . However,there we see something strange: K Rep ( φ ) sends the trivial representation to theprincipal series I = π ( ∅ , δ ∅ , I and the Steinberg representation of H ( R , q ). Theorem 5.23
The following are equivalent:1. G ( φ ) ⊗ id Q : G (cid:0) C ∗ r ( R , q ) (cid:1) ⊗ Q → G (cid:0) C ∗ r ( W ) (cid:1) ⊗ Q is a bijection2. K Rep ( φ ) ⊗ id Q : G (cid:0) C ∗ r ( W ) (cid:1) ⊗ Q → G (cid:0) C ∗ r ( R , q ) (cid:1) ⊗ Q is a bijection3. K ∗ ( φ ) ⊗ id Q : K ∗ (cid:0) C ∗ r ( W ) (cid:1) ⊗ Q → K ∗ (cid:0) C ∗ r ( R , q ) (cid:1) ⊗ Q is a bijection4. HP ∗ ( φ ) : HP ∗ ( S ( W )) → HP ∗ ( S ( R , q )) is a bijection5. HH ( φ ) : S ( W ) / [ S ( W ) , S ( W )] → S ( R , q ) / [ S ( R , q ) , S ( R , q )] is an isomor-phism of Fr´echet spacesProof. ⇔
2. By construction it suffices to show this for φ W t , for arbitrary W t ∈ T u /W . But φ W t is just a homomorphism between finite dimensionalsemisimple algebras, so K ( φ ) ⊗ id Q and G ( φ ) ⊗ id Q are linear maps betweenfinite dimensional vector spaces. With respect to the bases formed by irreduciblerepresentations the matrices of these two maps are each others transpose. Inparticular one of them is bijective if and only if the other is.2 ⇔
3. Consider the projectionpr : Ξ u / W → T u /W pr ( W ( P, W P r, δ, t )) = W r u t (5.76)With this map we make C ∗ r ( R , q )) into a C ( T u /W )-algebra. By (5.50) φ is C ( T u /W )-linear. Triangulate T u /W such that every subset T Gu /W with G ⊂ W becomes a subcomplex. In view of Proposition 2.21, 2 implies 3.62 Chapter 5. Parameter deformations in affine Hecke algebrasContrarily, suppose that K Rep ( φ ) ⊗ id Q is not surjective. By definition thereis a t ∈ T u such that K ( φ W t ) ⊗ id Q is not surjective. However the canonicalmap K ( C ∗ r ( R , q )) → K (cid:0) C ∗ r ( R , q ) /J cW t T rs (cid:1) is always surjective. This can be seen as follows. Every component of Ξ u intersectspr − ( W t ) in at most one W -orbit. If [ p ] ∈ K (cid:0) C ∗ r ( R , q ) /J cW t T rs (cid:1) then [ p ] ∈ K ( C ∗ r ( R , q )) maps to [ p ] if and only if the value of p on Ξ u ∩ pr − ( W t ) is asprescribed by p . It follows from Theorem 3.25 that such a p can always can befound. This shows that K ∗ ( φ ) ⊗ id Q and K ∗ ( φ ) are not surjective.Now suppose that K Rep ( φ ) ⊗ id Q is not injective. Pick W t such that K ( φ W t ) is not injective, with | W ,t | minimal for this property. Next pick V, V (cid:48) ∈ Rep W t ( C ∗ r ( W )) such that [ V ] − [ V (cid:48) ] ∈ ker K (cid:0) φ W t (cid:1) . Put T (cid:48) := { t ∈ T u : W ,t (cid:54)⊂ W ,t } and introduce the ideals I := J cT (cid:48) ⊂ C ∗ r ( W ) I := J cT (cid:48) T rs ⊂ C ∗ r ( R , q )Note that φ ( I ) ⊂ I . Recall the description C ∗ r ( W ) ∼ = C (cid:0) T u ; End C [ W ] (cid:1) W (5.77)from Lemma A.3, in combination with (2.105) and Theorem 3.15. These showthat it is possible to find m, n ∈ N and projections p, p (cid:48) ∈ M n ( I +0 ) such that p ( t )and p (cid:48) ( t ) yield the W ,t -modules mV and mV (cid:48) , for all t ∈ T W ,t u \ T (cid:48) . Now weinsert [ p ] − [ p (cid:48) ] in the commutative diagram K ( I ) → K ( C ∗ r ( W )) ↓ ↓ K ( I ) → K ( C ∗ r ( R , q )By assumption [ φ ( p )] − [ φ ( p (cid:48) )] = 0 ∈ K ( I )On the other hand, [ p ] and [ p (cid:48) ] are different on T W ,t u , so by Theorem 2.22 Ch W (cid:0) [ p ] − [ p (cid:48) ] (cid:1) (cid:54) = 0 ∈ ˇ H ∗ (cid:0)(cid:102) T u /W ; C (cid:1) Therefore K ( φ ) ⊗ Q and K ( φ ) are not injective.3 ⇔ ⇔
5. The localization of HH ( S ( R , q )) = S ( R , q ) / [ S ( R , q ) , S ( R , q )]at W ξ ∈ Ξ u / W is a complex vector space whose dimension is the number ofinequivalent irreducible constituents of π ( ξ ). This gives a fibration of Ξ u / W , and.4. K -theoretic conjectures 163by Theorem 3.25 HH ( S ( R , q )) can be regarded as the set of global sections of thesheaf F q of smooth sections of this fibration. The direct image of F q under (5.76) isthe sheaf of smooth sections of a fibration of T u /W . The fiber at W t is a vectorspace whose dimension is the number of irreducibles in Rep W t T rs ( C ∗ r ( R , q )).The Fr´echet space HH ( S ( W )) admits a similar description it terms of a sheaf F , but with fibers of dimension the number of irreducibles in Rep W t ( C ∗ r ( W )).Now φ induces a morphism F ( φ ) : F → pr ∗ ( F q )of sheaves over T u /W . If G ( φ ) ⊗ id Q is not bijective, then F and pr ∗ ( F q ) havedifferent stalks, so F ( φ ) and HH ( φ ) cannot be isomorphisms. On the otherhand, if G ( φ ) ⊗ id Q is bijective, then F ( φ )( W t ) : F ( W t ) → pr ∗ ( F q )( W t )is a bijection for every W t ∈ T u /W . This implies that F ( φ ) is an isomor-phism, see e.g. [47, Section II.1.6]. In particular HH ( φ ) = F ( φ )( T u /W ) is anisomorphism of Fr´echet spaces. (cid:50) Conjecture 5.24
The equivalent statements of Theorem 5.23 hold for every rootdatum and every positive label function. If q is an equal label function then by the Kazhdan-Lusztig classification (seepage 132) and by Theorem 5.3 none of the vector spaces in Theorem 5.23 dependson q , up to natural isomorphisms. It is probable, though not a priori certain, thatthese isomorphisms can be realized with φ .Actually, even for unequal labels strange things happen if Conjecture 5.24 doesnot hold. Suppose for example that HH ( φ ) is not injective. In that case therewould exist an x ∈ S ( W ) \ [ S ( W ) , S ( W )] such that φ ( x ) ∈ [ S ( R , q ) , S ( R , q )]However, since φ (cid:15) is an isomorphism ∀ (cid:15) >
0, we would have φ − (cid:15) φ ( x ) ∈ [ S ( R , q (cid:15) ) , S ( R , q (cid:15) )] ∀ (cid:15) > (cid:15) → φ − (cid:15) φ ( x ) and (cid:15) → z · q (cid:15) y − y · q (cid:15) z are both continuous on [0 , , ∀ x, y, z ∈ S ( R ). Maybe one can show that it isoutright impossible, by a thorough study of conjugacy classes in affine Weyl groups.Related results for finite Coxeter groups can be found in [46, Sections 3.2 and 8.2].Or suppose that G ( φ ) ⊗ id Q is not injective. Then there would exist t ∈ T u and( π, V ) , ( π (cid:48) , V (cid:48) ) ∈ Rep W t T rs ( C ∗ r ( R , q )) such that π ◦ φ and π (cid:48) ◦ φ are equivalent C ∗ r ( W )-representations. The Euler-Poincar´e pairing from (3.74) can help in this64 Chapter 5. Parameter deformations in affine Hecke algebrassituation. Assume for the moment that R is semisimple. By Theorem 5.2 thefinite dimensional semisimple subalgebras H ( R , I, q ) are rigid under q → q (cid:15) , so π (cid:12)(cid:12) H ( R ,I,q ) ∼ = ( π ◦ φ ) (cid:12)(cid:12) H ( R ,I,q ) ∼ = ( π (cid:48) ◦ φ ) (cid:12)(cid:12) H ( R ,I,q ) ∼ = π (cid:48) (cid:12)(cid:12) H ( R ,I,q ) (5.79)Therefore P n ( V ) Ω ∼ = P n ( V (cid:48) ) Ω for all n , and by Corollary 3.11Eul [ V ] = Eul [ V (cid:48) ] ∈ K ( H )Together with (3.77) this shows that [ V ] − [ V (cid:48) ] is in the radical of the Euler-Poincar´e pairing. However, we noticed on page 79 that the radical of EP is verylarge, so this does certainly not imply that π and π (cid:48) are equivalent. The nexttheorem is very useful to overcome this problem. It is analogous to results ofMeyer [91, Theorems 21 and 38] for Schwartz algebras of reductive p -adic groups. Theorem 5.25
Let R be any root datum and V, V (cid:48) ∈ Rep( S ( R , q )) . Then (cid:0) S ( R , q ) ⊗ H ( R ,q ) P ∗ ( V ) Ω , id S ( R ,q ) ⊗ d ∗ (cid:1) (5.80) is a finitely generated resolution of V . This resolution consists of projective mod-ules if R is semisimple. Moreover for such root data there are natural isomor-phisms Ext n H ( R ,q ) ( V, V (cid:48) ) ∼ = Ext n S ( R ,q ) ( V, V (cid:48) ) n ∈ N Proof.
This has been proved recently by Opdam and the author. We can constructa suitable contracting homotopy operator of the differential complex (cid:0) P ∗ ( V ) Ω , d ∗ (cid:1) Using the temperedness of V we can show that this operator extends continuouslyto the complex (5.80). The details will be published elsewhere. (cid:50) .Suppose that both V and V (cid:48) are direct sums of discrete series representations.By Theorem 3.25 a discrete series module is projective in Rep ( S ( R , q )), so withTheorem 5.25 EP (cid:0) [ V ] − [ V (cid:48) ] , [ V ] − [ V (cid:48) ] (cid:1) = ∞ (cid:80) n =0 ( − n dim Ext n S ( R ,q ) (cid:0) [ V ] − [ V (cid:48) ] , [ V ] − [ V (cid:48) ] (cid:1) = dim Hom S ( R ,q ) (cid:0) [ V ] − [ V (cid:48) ] , [ V ] − [ V (cid:48) ] (cid:1) (5.81)Clearly this is positive whenever V and V (cid:48) are inequivalent. This does not onlyimply π ◦ φ (cid:54)∼ = π (cid:48) ◦ φ , but also that G ( φ ) (cid:0) [ V ] − [ V (cid:48) ] (cid:1) is not in the radical of EP. (5.82)Hence this element cannot be written as a sum of virtual representations that areinduced from proper parabolic subalgebras. Because every irreducible tempered.4. K -theoretic conjectures 165representation is a direct summand of a representation that is parabolically in-duced from a discrete series representation, this is an essential part of Conjecture5.24.In some important cases this actually suffices to prove the conjecture. Namely,using the theory of R-groups [40, 99] it can be shown that for certain labelledroot data all representations of the form π ( P, δ, t ) with (
P, δ, t ) ∈ Ξ u are irre-ducible. Using Theorems 3.19 and 5.25 we can apply an inductive argument toverify Conjecture 5.24 in such cases. See Section 6.7 for more details.Let us have another look at Theorem 5.23. Clearly the first three statementscan also be formulated with integral coefficients: Proposition 5.26
The following are equivalent:1. G ( φ ) : G (cid:0) C ∗ r ( R , q ) (cid:1) → G (cid:0) C ∗ r ( W ) (cid:1) is an isomorphism2. K Rep ( φ ) : G (cid:0) C ∗ r ( W ) (cid:1) → G (cid:0) C ∗ r ( R , q ) (cid:1) is an isomorphism3. K ∗ ( φ ) : K ∗ (cid:0) C ∗ r ( W ) (cid:1) → K ∗ (cid:0) C ∗ r ( R , q ) (cid:1) is an isomorphismProof. This is completely analogous to the corresponding part of the proof ofTheorem 5.23. (cid:50)
One of the motivations for considering these maps is that K ∗ ( φ ) is natural, inthe sense that it can be constructed without really using φ . The idea is that smallperturbations of invertibles or idempotents have no effect on classes in K -theory.By Theorem 2.27 K ∗ ( S ( W )) is finitely generated. Using Theorem 2.12 we canfind k ∈ N and a finite set of idempotents and invertibles in M k ( S ( W )) whichgenerates K ∗ ( S ( W )). Let ( u, q ) ∈ M k ( S ( R , q )) be such an invertible. In viewof Proposition 5.9 and the remark on page 146 there exists an (cid:15) u > u, q (cid:15) ) ∈ GL k ( S ( R , q (cid:15) )) ∀ (cid:15) < (cid:15) u To handle idempotents in a similar way we need holomorphic functional calculus.Define the holomorphic function f p on { z ∈ C : (cid:60) ( z ) (cid:54) = 1 / } by f p ( z ) = (cid:26) (cid:60) ( z ) > /
20 if (cid:60) ( z ) < / f p ( x ) is idempotent whenever it is defined. Let ( e, q ) ∈ M k ( S ( R , q ))be an idempotent. It is clear from Proposition 5.6 that ∃ (cid:15) e > e, q (cid:15) ) − / − ai ∈ GL k ( S ( R , q (cid:15) )) ∀ (cid:15) < (cid:15) e , ∀ a ∈ R In fact, by direct calculation one can show that this holds for all (cid:15) such that (cid:13)(cid:13) λ ( e, q (cid:15) ) − λ ( e, q ) (cid:13)(cid:13) B ( H ( R )) <
12 + 4 (cid:13)(cid:13) λ ( e, q ) (cid:13)(cid:13) B ( H ( R )) For such (cid:15) the idempotent f p ( e, q (cid:15) ) is well-defined.66 Chapter 5. Parameter deformations in affine Hecke algebras Lemma 5.27
The following equalities of K -theory classes hold for u and e asabove. [( u, q (cid:15) )] = K ( φ − (cid:15) φ )[( u, q )] ∈ K ( S ( R , q (cid:15) )) ∀ (cid:15) ∈ (0 , (cid:15) u )[ f p ( e, q (cid:15) )] = K ( φ − (cid:15) φ )[( e, q )] ∈ K ( S ( R , q (cid:15) )) ∀ (cid:15) ∈ (0 , (cid:15) e ) Proof.
For any x ∈ M k ( C ) ⊗ S ( R ) the map[0 , → M k ( C ) ⊗ S ( R ) : (cid:15) → φ − (cid:15) φ ( x )is continuous. Hence ( u, q (cid:15) ) and φ − (cid:15) φ ( u, q ) are homotopic in GL k ( S ( R , q (cid:15) )),for some small (cid:15) >
0. Clearly this implies that φ − (cid:15) φ (cid:15) ( u, q (cid:15) ) and φ − (cid:15) φ ( u, q )are homotopic ∀ (cid:15) ∈ (0 , (cid:15) u ). But there also is a path from φ − (cid:15) φ (cid:15) ( u, q (cid:15) ) to ( u, q (cid:15) )along elements of the form φ − (cid:15) φ (cid:15) ( u, q (cid:15) ).Similarly, by Corollary 5.10[0 , (cid:15) e ) → M k ( C ) ⊗ S ( R ) : (cid:15) → f p ( e, q (cid:15) )is continuous. According to [10, Proposition 4.3.2] there is a small (cid:15) > f p ( e, q (cid:15) ) and φ − (cid:15) φ ( e, q ) are homotopic in Idem( M k ( S ( R , q (cid:15) )). But then, asabove for u, φ − (cid:15) φ ( e, q ) and f p ( e, q (cid:15) ) are homotopic via φ − (cid:15) φ (cid:15) ( f p ( e, q (cid:15) )) . (cid:50) So we have a family of pre- C ∗ -algebras S ( R , q )) which are independent of q asFr´echet spaces, and whose multiplication depends continuously on q . Moreover,replacing q by q (cid:15) with (cid:15) > K ∗ ( φ ) can be constructed without using φ . Therefore itis not unreasonable to suspect the following. Conjecture 5.28
For any root datum R and positive label function q the map K ∗ ( φ ) : K ∗ ( S ( W )) → K ∗ ( S ( R , q )) is an isomorphism. As mentioned, this conjecture stems from Baum, Connes and Higson [5], atleast in the equal label case. Independently, Opdam [98, p. 533] stated it forunequal labels. In Chapter 6 we will verify Conjecture 5.28 for some classical rootdata.Consider the root datum
R × Z with the unique label function that extends q .By (3.90) S ( R × Z , q ) ∼ = S ( R , q ) (cid:98) ⊗S ( Z ) ∼ = C ∞ (cid:0) S ; S ( R , q ) (cid:1) (5.83)In Lemma 2.17 we constructed natural isomorphisms K ( S ( R × Z , q )) ∼ ←−− K ∗ ( S ( R , q )) ∼ −−→ K ( S ( R × Z , q )) (5.84)Therefore it suffices to prove Conjecture 5.24 either for K and every ( R , q ), orfor K and every ( R , q ). Probably the K -case is easier, for two reasons. Firstly,.4. K -theoretic conjectures 167invertibles are more flexible than idempotents. If we perturb them a little theyremain invertible, so we can do without holomorphic functional calculus. Secondly,we can find a bound, uniform in q , on the size of the matrices that we need torepresent all K -classes. In fact, by Proposition 2.16 we can bound the topologicalstable rank by tsr ( C ∗ r ( R , q )) ≤ | W | (cid:0) (cid:98) dim T u / (cid:99) (cid:1) (5.85)Now Theorems 2.13 and 2.15 show that K ( S ( R , q )) ∼ = π (cid:0) GL n ( S ( R , q )) (cid:1) ∀ n ≥ | W | (1 + rk( X ) /
2) (5.86)One way to attack Conjecture 5.28 goes approximately as follows. Pick a finiteset of generators of K ( S ( R , q )). Find suitable representants u i ∈ GL n ( S ( R , q )),i.e. u i should lie in M n ( C ) ⊗ span { N w : N ( w ) ≤ M } with M ”small” and sp( u i ) should lie in a ”small” neighborhood of the unit circlein C . If q is close to q one may hope that to every u i one can associate in anunambiguous way a unique homotopy class in GL n ( S ( W )). This class should beconstructed by applying the isomorphisms φ − (cid:15) and by perturbing u i a little. Inparticular it should contain elements of M n ( C ) ⊗ S ( R ) that are homotopic to u i in GL n ( S ( R , q )).An analogue of Conjecture 5.28 does hold for noncommutative tori. Let T n = R n / Z n ∼ = ( S ) n be the standard compact n -dimensional torus. In this settingthe underlying group W becomes Z n , and q is replaced by a skew-symmetricbilinear form θ on Z n . By taking iterated crossed products with Z , one constructsa C ∗ -algebra which is a deformation of C ( T n ) and is commonly denoted by A θ . Ithas a holomorphically closed dense subalgebra S ( Z n , θ ) which, as a Fr´echet space,is naturally isomorphic to S ( Z n ). By deforming θ Elliott [43, Theorem 2.2] provedthat there exists a natural group isomorphism K ∗ ( A θ ) ∼ −−→ (cid:94) Z n (5.87)For θ = 0 this can be interpreted geometrically as the classical Chern character Ch : K ∗ ( T n ) ∼ −−→ H ∗ ( T n ; Z )However, there are also quite big differences between noncommutative tori andaffine Hecke algebras. Namely, the structure of A θ is very different for θ rationalor irrational, and an essential ingredient in Elliott’s proof is the Pimsner-Voiculescuexact sequence, which is not available for crossed products with non-cyclic groups.68 Chapter 5. Parameter deformations in affine Hecke algebras hapter 6 Examples and calculations
The final chapter of this book is completely different from the others. We barelystate or prove theorems here, we mostly make calculations.In Chapters 3 and 5 we studied affine Hecke algebras in a very abstract way,almost without mentioning any examples. However the results in those chapterscould hardly have been obtained without first checking, by hand, what happensin some simple exemplary cases. Basically we have two goals: we want have toexamples of all the objects we introduced in Chapter 3, and we want to verify theconjectures made in Section 5.4 in some cases.So we must devise a strategy to calculate the K -theory of the C ∗ -completion C ∗ r ( R , q ) of an affine Hecke algebra with root datum R and label function q , andto find the homomorphisms (hopefully isomorphisms) K ∗ ( φ ) : K ∗ (cid:0) C ∗ r ( R , q ) (cid:1) → K ∗ (cid:0) C ∗ r ( R , q ) (cid:1) (6.1)For q we can use (2.105) and (3.91), which say that K ∗ (cid:0) C ∗ r ( W ) (cid:1) ⊗ C ∼ = K ∗ ( C ( T u ) (cid:111) W ) ⊗ C ∼ = ˇ H ∗ (cid:0)(cid:102) T u ; C (cid:1) W ∼ = ˇ H ∗ (cid:0)(cid:102) T u (cid:14) W ; C (cid:1) (6.2)Moreover, if ˇ H ∗ (cid:0)(cid:102) T u (cid:14) W ; Z (cid:1) is torsion free, then by Theorem 2.24 K ∗ (cid:0) C ∗ r ( W ) (cid:1) = K ∗ (cid:16) C (cid:0) T u ; End C [ W ] (cid:1) W (cid:17) ∼ = ˇ H ∗ (cid:0)(cid:102) T u (cid:14) W ; Z (cid:1) (6.3)In general our procedure will involve the following steps.1. Explicitly write down the root datum and the associated Weyl groups.2. Determine the residual cosets and distinguish the different ”genericity classes”of label functions.For every q there is, as noticed in (5.40), a canonical decomposition C ∗ r ( R , q ) = (cid:77) P C ∗ r ( R , q ) P (6.4)where P runs over certain sets of simple roots.16970 Chapter 6. Examples and calculations3. List a good set of P ’s.For every chosen P we do the following:4. Determine the root datum R P
5. Determine the discrete series of H ( R P , q P ), and all the relevant intertwiningoperators.This is the only step where we can still encounter theoretical difficulties. Theproblem is that in general it is not known how many inequivalent discreteseries representations there are. Fortunately we can decide this in the caseswe consider.6. Describe C ∗ r ( R , q ) P and its primitive ideal spectrum.7. Calculate K ∗ (cid:0) C ∗ r ( R , q ) (cid:1) P .Our main tools for this are excision and Proposition 2.21. In principle thesewill always lead to the answer, but it is tedious work that becomes unprac-tical in higher dimensions. On the other hand things become easier if weforget about torsion elements, for then we can use sheaf cohomology. Forthis reason will sometimes be satisfied with K ∗ (cid:0) C ∗ r ( R , q ) P (cid:1) ⊗ Z C .8. Find generating idempotents and invertibles, as explicit as possible.9. Compare K ∗ (cid:0) C ∗ r ( R , q ) (cid:1) and K ∗ (cid:0) C ∗ r ( R , q ) (cid:1) .10. Determine K ∗ ( φ ) in terms of the given generators.The results of these calculations are • There are no examples for which (6.1) is known to be no isomorphism. • (6.1) is an isomorphism for some root data of low rank • (6.1) is an isomorphism the root data R ( GL n ) and R ( A n − ) ∨ .We will also see that these K -groups tend to be torsion free. This is due tothe fact that the primitive ideal spaces of affine Hecke algebras look like quotientsof tori by reflection groups, or direct products and unions of those. The root datawe study all have free abelian K -groups, but whether this holds in general is hardto say. A A playsan important role in the realm of Hecke algebras. Every Iwahori-Hecke algebra isin a sense built from such rank one Hecke algebras. There are two semisimple root.1. A R = A , corresponding to the root lattice and the weight lattice. Wewill study the associated affine Hecke algebras in detail, and show that Conjecture5.28 holds for these root data.The notation R ( A ) will be reserved for X the root lattice. We start with theother case. Thus we consider the root datum R ( A ) ∨ = ( X, Y, R , R ∨ , F ) with X = Z Q = 2 Z X + = Z ≥ Y = Q ∨ = Z T = C × R = R = {± α } = {± } R ∨ = R ∨ = (cid:8) ± α ∨ (cid:9) = {± } F = { α } W = { e, s α } s = s α : x → − x s = t s = t s t − : x → − xS aff = { s , s } W (cid:54) = W aff = (cid:104) s , s | s = s = e (cid:105) Ω = { e, ω } = { e, t s } For any label function q we have q ( s ) = q ( s ) = q α ∨ , and we denote this valuesimply by q . Then c α = (1 − q − θ − )(1 − θ − ) − so generically the residual points are q / , q − / , − q / , − q − / (6.5)Hence there are only two essentially different cases, depending on whether q equals1 or not. • group case q = From Theorem 3.15 we know that every irreducible representation is a direct sum-mand of a unitary principal series I t = π ( ∅ , δ ∅ , t ). The underlying vector space of I t is C [ W ] = C T e + C T s and the intertwiner π ( s , ∅ , δ ∅ , t ) : I t → I t − is simply right multiplication by T s .So with respect to the orthonormal basis (cid:8) − / ( T e + T s ) , − / ( T e − T s ) (cid:9) (6.6)we have S ( W ) ∼ = (cid:26) f ∈ C ∞ (cid:0) S ; M ( C ) (cid:1) : f ( t − ) = (cid:18) − (cid:19) f ( t ) (cid:18) − (cid:19)(cid:27) C ∗ r ( W ) ∼ = { f ∈ C (cid:0) [ − , M ( C ) (cid:1) : f ( ±
1) is diagonal } (6.7)72 Chapter 6. Examples and calculationsThe spectrum of these algebras is the non-Hausdorff spacePrim( C ∗ r ( W )) ∼ = (cid:99)(cid:99) (cid:99)(cid:99) To calculate the K -theory we use the extension0 → C ([ − , , { , } ; M ( C )) → C ∗ r ( W ) → C → −
1. The associated exact hexagon is0 → K ( C ∗ r ( W )) → Z ↑ ↓ ← K ( C ∗ r ( W )) ← Z By comparing this with the standard extension0 → C ([ − , , { , } ) → C ( S ) → C → K ( C ∗ r ( W )) ∼ = Z K ( C ∗ r ( W )) = 0 (6.8)We can even find explicit generating projections, namely p a = ( T e + T s ) / p b = ( T e − T s ) / p c = T e / − (( θ + θ − ) T s + i ( θ − − θ ) T e ) / p d = T e / θ + θ − ) T s + i ( θ − − θ ) T e ) / p a ] + [ p b ] = [ p c ] + [ p d ] = [1] ∈ K ( C ∗ r ( W )) • generic, equal label case q (cid:54) = • P = ∅ R P = ∅ R ∨ P = ∅ X P = X X P = 0 Y P = Y Y P = 0 T P = T T P = { } K P = { } W P = W ( P, P ) = W P P = W W P = { e } .1. A π ( s , ∅ , δ ∅ , t ) : I t → I t − and ı os = ( T s (1 − θ ) + ( q − θ )( q − θ ) − π ( s , ∅ , δ ∅ ,
1) = π ( s , ∅ , δ ∅ , −
1) = 1 C ∗ r ( R , q ) P ∼ = C (cid:0) [ − , M ( C ) (cid:1) Prim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = S /W ∼ = [ − , • P = { α } R P = R R ∨ P = R ∨ W P = W ( P, P ) = W P P = { e } W P = W X P = 0 X P = X Y P = 0 Y P = YT P = { } T P = T K P = { } From Proposition 3.20.2 we know that there is exactly one discrete series represen-tation for every orbit of residual points. So the spectrum of C ∗ r ( R , q ) contains twoisolated points, which we call δ and δ − , by the sign of their central character. C ∗ r ( R , q ) P ∼ = C By brute calculation one finds the associated projectors p = (cid:80) w ∈ W aff ( − q ) (cid:96) ( w ) T w ( T e − T ω ) (cid:16) (cid:80) w ∈ W q ( w ) − (cid:17) − if q > p = (cid:80) w ∈ W T w (cid:16) (cid:80) w ∈ W q ( w ) − (cid:17) − if q < p − = (cid:80) w ∈ W ( − q ) (cid:96) ( w ) T w (cid:16) (cid:80) w ∈ W q ( w ) (cid:17) − if q > p − = (cid:80) w ∈ W aff T w ( T e − T ω ) (cid:16) (cid:80) w ∈ W q ( w ) (cid:17) − if q < P = ∅ and P = { α } , the spectrum of C ∗ r ( R , q ) becomesthe Hausdorff spacePrim( C ∗ r ( W )) ∼ = (cid:115) (cid:115) This implies K ( C ∗ r ( R , q )) ∼ = Z K ( C ∗ r ( R , q )) = 0 (6.11)Let p be any rank one projector in C ([ − , M ( C )). Then (the classes of) p − , p and p generate K ( C ∗ r ( R , q )), and we have[ p − ] + [ p ] + 2[ p ] = [1] ∈ K ( C ∗ r ( R , q ))Now we can compare the cases q = 1 and q (cid:54) = 1. Looking carefully at the behaviournear t = 1 and t = −
1, we find that K ( φ ) is always an isomorphism. We describethis map completely with the following table.74 Chapter 6. Examples and calculations q < q = 1 q > p + p + p − p a p p p b p + p + p − p + p − p c p + p p + p p d p + p − (6.12)Let us move on to the type A Hecke algebras with X the root lattice. This meansthat we work with the root datum R ( A ): X = Q = Z X + = Z ≥ Y = Z Q ∨ = 2 Z T = C × R = {± α } = {± } R = {± } R ∨ = {± α ∨ } = {± } R ∨ = {± } F = { α } W = { e, s α } s = s α : x → − x s = t s : x → − xS aff = { s , s } W = W aff = (cid:104) s , s | s = s = e (cid:105) Now s and s are no longer conjugate, so q = q ( s ) and q = q ( s ) may bedifferent. By definition q α ∨ = q q α ∨ / = q q − c α = (1 + q − / q / θ − )(1 − q − / q − / θ − )(1 − θ − ) − Generically the residual points are q / q / q − / q − / − q / q − / − q − / q / (6.13)From this we see that there are four cases to study: the group case q = q = 1,the equal label case q = q (cid:54) = 1, the ”inverse label” case q = q − (cid:54) = 1 and thegeneric case. • group case q = q = This is the same as the group case for X equal to the weight lattice of A . • equal label case q = q = q (cid:54) = • P = ∅ R P = ∅ R ∨ P = ∅ X P = X X P = 0 Y P = Y Y P = 0 T P = T T P = { } K P = { } W P = W ( P, P ) = W P P = W W P = { e } ı os = ( T s (1 − θ ) + ( q − θ )( q − θ ) − .1. A (cid:8) ( T e + T s )(1 + q ) − / , ( qT e − T s )( q + q ) − / (cid:9) (6.14)of H ( W , q ) we have π ( s , ∅ , δ ∅ ,
1) = (cid:18) (cid:19) π ( s , ∅ , δ ∅ , −
1) = (cid:18) − (cid:19) C ∗ r ( R , q ) P ∼ = (cid:8) f ∈ C (cid:0) [ − , M ( C ) (cid:1) : f ( −
1) is diagonal (cid:9)
Prim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = (cid:99)(cid:99) • P = { α } R P = R R ∨ P = R ∨ X P = 0 X P = X Y P = 0 Y P = YT P = { } T P = T K P = { } W P = W ( P, P ) = W P P = { e } W P = W Obviously − q / q − / = − q − / q / = −
1, so these points are not residual forthis particular label function. On the other hand, by Proposition 3.20.2 thereis a unique discrete series representation δ with central character q ± . It hasdimension 1, and the corresponding projection is p = (cid:80) w ∈ W ( − q ) (cid:96) ( w ) T w (cid:16) (cid:80) w ∈ W q ( w ) − (cid:17) − if q > p = (cid:80) w ∈ W T w (cid:16) (cid:80) w ∈ W q ( w ) (cid:17) − if q < C ∗ r ( R , q ) ∼ = (cid:8) f ∈ C (cid:0) [ − , M ( C ) (cid:1) : f ( −
1) is diagonal (cid:9) ⊕ C Prim (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = (cid:99)(cid:99) (cid:115) Evaluating at − q ± yields an extension0 → C (cid:0) [ − , , {− } ; M ( C ) (cid:1) → C ∗ r ( R , q ) → C → K -theory is0 → K ( C ∗ r ( R , q )) → Z ↑ ↓ ← K ( C ∗ r ( R , q )) ←
076 Chapter 6. Examples and calculationsThis shows that K ( C ∗ r ( R , q )) ∼ = Z K ( C ∗ r ( R , q )) = 0 (6.16)Generating projections are p = (cid:18)(cid:18) (cid:19) , (cid:19) p a = (cid:18)(cid:18) (cid:19) , (cid:19) p b = (cid:18)(cid:18) (cid:19) , (cid:19) Explicitly this works out to p a = ( T e + T s )(1 + q ) − if q > p a = ( T e + T s )(1 + q ) − − p if q < p b = ( qT e − T s )(1 + q ) − − p if q > p a = ( T e + T s )(1 + q ) − if q < • inverse label case q = q = q − (cid:54) = • P = ∅ R P = ∅ R ∨ P = ∅ X P = X X P = 0 Y P = Y Y P = 0 T P = T T P = { } K P = { } W P = W ( P, P ) = W P P = W W P = { e } ı os = ( T s (1 + θ ) + (1 − q ) θ )( q + θ ) − With respect to the basis (6.14) we have π ( s , ∅ , δ ∅ ,
1) = (cid:18) − (cid:19) π ( s , ∅ , δ ∅ , −
1) = (cid:18) (cid:19) C ∗ r ( R , q ) P ∼ = (cid:8) f ∈ C (cid:0) [ − , M ( C ) (cid:1) : f (1) is diagonal (cid:9) Prim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = (cid:99)(cid:99) • P = { α } R P = R R ∨ P = R ∨ X P = 0 X P = X Y P = 0 Y P = YT P = { } T P = T K P = { } W P = W ( P, P ) = W P P = { e } W P = W Now we have q / q / = q − / q − / = 1, so these points are not residual. There is aunique discrete series representation δ − with central character − q ± . Its projectoris already a little more difficult to describe. For w ∈ W write (cid:96) ( w ) = (cid:96) ( w )+ (cid:96) ( w ),where (cid:96) counts the number of factors s i in an reduced expression for w . Notice.1. A s and s in W aff . p − = (cid:80) w ∈ W ( − q ) (cid:96) ( w ) / T w (cid:16) (cid:80) w ∈ W q − (cid:96) ( w ) (cid:17) − if q > p − = (cid:80) w ∈ W ( − q ) (cid:96) ( w ) / T w (cid:16) (cid:80) w ∈ W q (cid:96) ( w ) (cid:17) − if q < C ∗ r ( R , q ) ∼ = C ⊕ (cid:8) f ∈ C (cid:0) [ − , M ( C ) (cid:1) : f ( −
1) is diagonal (cid:9)
Prim (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = (cid:99)(cid:99)(cid:115) Like in the equal case this leads to K ( C ∗ r ( R , q )) ∼ = Z K ( C ∗ r ( R , q )) = 0 (6.19)and generating projections are p − = (cid:18) , (cid:18) (cid:19)(cid:19) p a = (cid:18) , (cid:18) (cid:19)(cid:19) p b = (cid:18) , (cid:18) (cid:19)(cid:19) Now they are given by p a = ( T e + T s )(1 + q ) − if q > p a = ( T e + T s )(1 + q ) − − p − if q < p b = ( qT e − T s )(1 + q ) − − p − if q > p b = ( qT e − T s )(1 + q ) − if q < • generic case q (cid:54) = q (cid:54) = q − • P = ∅ R P = ∅ R ∨ P = ∅ X P = X X P = 0 Y P = Y Y P = 0 T P = T T P = { } K P = { } W P = W ( P, P ) = W P P = W W P = { e } ı os = ( T s (1 + θ ) + (1 − q ) θ )( q + θ ) − In this case all unitary principal series representations are irreducible: π ( s , ∅ , δ ∅ ,
1) = 1 π ( s , ∅ , δ ∅ , −
1) = 1 C ∗ r ( R , q ) P ∼ = C (cid:0) [ − , M ( C ) (cid:1) Prim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = S /W ∼ = [ − , • P = { α } R P = R R ∨ P = R ∨ X P = 0 X P = X Y P = 0 Y P = YT P = { } T P = T K P = { } W P = W ( P, P ) = W P P = { e } W P = W The residual points were already listed in (6.13). By Proposition 3.20.2 there areexactly two inequivalent discrete series representations, δ and δ − . C ∗ r ( R , q ) P ∼ = C To write down the corresponding projections we have to distinguish four cases. p = (cid:88) w ∈ W ( − (cid:96) ( w ) q ( w ) − T w (cid:16) (cid:88) w ∈ W q ( w ) − (cid:17) − if q > q − (6.21) p = (cid:88) w ∈ W T w (cid:16) (cid:88) w ∈ W q ( w ) (cid:17) − if q < q − (6.22) p − = (cid:88) w ∈ W ( − q ) (cid:96) ( w ) / T w (cid:16) (cid:88) w ∈ W q − (cid:96) ( w )1 q (cid:96) ( w )0 (cid:17) − if q < q (6.23) p − = (cid:88) w ∈ W ( − q ) (cid:96) ( w ) / T w (cid:16) (cid:88) w ∈ W q − (cid:96) ( w )0 q (cid:96) ( w )1 (cid:17) − if q > q (6.24)If we consider δ and δ − only as representations of H ( W , q ), then in this list(6.21) and (6.23) are deformations of the sign representation, while (6.22) and(6.24) are deformations of the trivial representation.We conclude that C ∗ r ( R , q ) ∼ = C ⊕ C (cid:0) [ − , M ( C ) (cid:1) ⊕ C Prim (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = (cid:115) (cid:115) Hence in the generic case also K ( C ∗ r ( R , q )) ∼ = Z K ( C ∗ r ( R , q )) = 0 (6.25)Canonical generators are [ p − ] , [ p ] and [ p ], where p is any rank one projectorin C (cid:0) [ − , M ( C ) (cid:1) .So for this root datum the K -groups are independent of the label function. More-over the various maps K ( φ ) all turn out to be isomorphisms. We list the imagesof the projections p a , p b , p c and p d below, in that order..2. GL q = q < q − < q > q q = q > p a p p a p b + p − p + p + p − p b + p p b p + p p a + p p a + p − p + p − p b q − > q < q q = q = 1 q − < q > q p + p p a p + p − p + p − p b p + p p p c p + p + p − p + p + p − p d p q = q < q − > q < q q − = q < p a + p p + p + p − p a + p − p b p p b p a p + p − p b + p − p b + p p + p p a (6.26) GL R ( GL ). X = Z Q = { ( n, − n ) : n ∈ Z } X + = { ( m, n ) ∈ Z : m ≥ n } Y = Z Q ∨ = { ( n, − n ) : n ∈ Z } T = ( C × ) t = ( t , t ) = ( t (1 , , t (0 , R = {± α } = {± (1 , − } = R R ∨ = {± α ∨ } = {± (1 , − } = R ∨ F = α W = { e, s α } s = s α : ( m, n ) → ( n, m ) s = t α s α = t (1 , s t ( − , : ( m, n ) → ( n + 1 , m − S aff = { s , s } W (cid:54) = W aff = (cid:104) s , s | s = s = e (cid:105) Ω = (cid:104) ω (cid:105) = (cid:104) t (1 , s (cid:105) ∼ = Z For any label function q we have q ( s ) = q ( s ) = q α ∨ , so we call this value q .There are no residual point because R is not semisimple. We do have two residualcosets of dimension one, namely { t ∈ T : t − t = q } and { t ∈ T : t − t = q − } So there are only two really different cases, q = 1 and q (cid:54) = 1. • group case q =
80 Chapter 6. Examples and calculationsAs said before, we only need to look at unitary principal series representations.There is a single nonscalar intertwiner π ( s , ∅ , δ ∅ , t ) : I ( t ,t ) → I ( t ,t ) It is givenby right multiplication with T s , so with respect to the basis (6.6) we have S ( W ) ∼ = (cid:26) f ∈ C ∞ (cid:0) T ; M ( C ) (cid:1) : f ( t , t ) = (cid:18) − (cid:19) f ( t , t ) (cid:18) − (cid:19)(cid:27) Let M be the closed M¨obius strip and ∂M its boundary. We see that C ∗ r ( W ) ∼ = { f ∈ C ( M ; M ( C )) : f ( m ) is diagonal if m ∈ ∂M } Consider the ideal A = { f ∈ C ( M ; M ( C )) : f ( m ) = (cid:18) ∗
00 0 (cid:19) if m ∈ ∂M } According to Proposition 2.21 the inclusion A → C ( M ; M ( C )) induces an iso-morphism on K -theory. However, M is homotopy equivalent to a circle, so K ( A ) ∼ = K ( A ) ∼ = Z Moreover ∂M ∼ = S , so from0 → A → C ∗ r ( W ) → C ( ∂M ) → Z → K ( C ∗ r ( W )) → Z ↑ ↓ Z ← K ( C ∗ r ( W )) ← Z Now it is not difficult to see that the upper part of this hexagon is exact, andhence K (cid:0) C ∗ r ( W ) (cid:1) ∼ = Z K (cid:0) C ∗ r ( W ) (cid:1) ∼ = Z (6.27)Generating projections and unitaries are p a = ( T e + T s ) / p b = ( T e − T s ) / θ (0 , u = θ (0 , p a + θ (0 , − p b (6.28) • generic, equal label case q (cid:54) = • P = ∅ .2. GL R P = ∅ R ∨ P = ∅ X P = X X P = 0 Y P = Y Y P = 0 T P = T T P = { } K P = { } W P = W ( P, P ) = W P P = W W P = { e } ı os = ( T s (1 − θ (1 , − ) + (1 − q ) θ (1 , − )( q + θ (1 , − ) − If s ( t ) = t then π ( s , ∅ , δ ∅ , t ) = 1, so C ∗ r ( R , q ) P ∼ = C ( M ; M ( C ))Prim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = T u /W ∼ = M • P = { α } R P = R R ∨ P = R P X P = X/ Z α ∼ = Z X P = X/ ( R ∨ P ) ⊥ ∼ = Z α/ Y P = Y ∩ R ⊥ P = Z (1 , Y P = Y ∩ Q R ∨ P = Z α ∨ T P = { ( t , t ) : t ∈ C × } T P = { ( t , t − ) : t ∈ C × } K P = { (1 , , ( − , − } = { , k P } The root datum R P is isomorphic to R ( A ) ∨ , so we can use the description of thediscrete series on page 175. The representations π ( P, δ , ( t , t )) and π ( P, δ , ( − t , − t )) are intertwined by π ( k P ). C ∗ r ( R , q ) P ∼ = C (cid:0) S (cid:1) Prim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = (cid:0) P, W P ( q / , q − / ) , δ , T Pu (cid:1) ∼ = S The associated projector is p α = (cid:80) w ∈ W aff ( − q ) (cid:96) ( w ) T w (cid:16) (cid:80) w ∈ W aff q ( w ) − (cid:17) − if q > p α = (cid:80) w ∈ W aff T w (cid:16) (cid:80) w ∈ W aff q ( w ) (cid:17) − if q < K (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = Z (6.30)These groups are generated by the classes of the projections p α = (cid:18)(cid:18) (cid:19) , (cid:19) and p = (cid:18)(cid:18) (cid:19) , (cid:19) and of the invertibles u α = (cid:18)(cid:18) (cid:19) , id S (cid:19) and u = (cid:18)(cid:18) r
00 1 (cid:19) , (cid:19)
82 Chapter 6. Examples and calculationswhere r : M → S is a homotopy equivalence. Explicitly we may take u α = p α θ (1 , + 1 − p α u = θ (1 , (1 − p α ) + p α p = ( T e + T s )(1 + q ) − if q > p = ( T s − qT e )(1 + q ) − if q < K -theory of S ( R , q )) is independent of q . Thegroup isomorphisms K ( φ ) are as follows: q < q = 1 q > p + p α p a p p p b p + p α u θ (1 , u u u α u u u − α (6.32) A A as embedded in R , but only as atwodimensional object. There are two semisimple root data with R of type A ,depending on whether X is the root lattice or the weight lattice. The latter caseis easier, so let us draw this root system together with the fundamental weights x and x . x x -a b g a-b-g .3. A R ( A ) ∨ is described by X = Z x + Z x = Z (cid:0) , − / (cid:1) + Z (cid:0) / , − / / (cid:1) X + = N x + N x Q = Z (1 ,
0) + Z (cid:0) − / , √ / (cid:1) Y = Q ∨ = Z (2 ,
0) + Z (cid:0) , √ (cid:1) T = ( C × ) t = ( t , t ) = ( t ( x ) , t ( x )) R = {± α, ± β, ± γ } = {± (1 , , ± (cid:0) − / , √ / (cid:1) , ± (cid:0) / , √ / (cid:1) } = R R ∨ = (cid:8) ± α ∨ , ± β ∨ , ± γ ∨ (cid:9) = {± (1 , , ± (cid:0) − / , √ / (cid:1) , ± (cid:0) / , √ / (cid:1) } = R ∨ F = { α, β } W = (cid:104) s α , s β | s α = s β = ( s α s β ) = e (cid:105) ∼ = S s = s α : ( n, m ) → ( − n, m ) s = s β : (cid:0) n + m/ , m √ / (cid:1) → (cid:0) m + n/ , n √ / (cid:1) s = t γ s γ : (cid:0) n + m/ , m √ / (cid:1) → (cid:0) (1 + n − m ) / , (1 − n − m ) √ / (cid:1) S aff = { s , s , s } W (cid:54) = W aff = (cid:104) s , W | s = ( s s ) = ( s s ) = e (cid:105) Ω = { e, ω , ω } = { e, t x s β s α , t x s α s β } For any label function q we have q ( s ) = q ( s ) = q ( s ) = q ( s ) = q α ∨ = q β ∨ = q γ ∨ so we denote this value simply by q . c η = (1 − q − θ − η )(1 − θ − η ) − for η ∈ { α, β, γ } . Generically there are 6 tempered residual circles and 18 residualpoints. Representatives for the W -conjugacy classes are( q, q ) ( qζ, qζ ) ( qζ , qζ ) and (cid:8) ( q / t , q / t ) : t ∈ T (cid:9) (6.33)where ζ = e πi/ is a root of unity. • group case q = In the compact torus T u there are three W -invariant points:(1 ,
1) ( ζ, ζ ) ( ζ , ζ ) (6.34)Furthermore we have the following circles with nontrivial stabilizers: { ( t , t ) : t ∈ T } s α { ( t , t ) : t ∈ T } s β { ( t , t − ) : t ∈ T } s γ These circles are conjugate under W . Therefore Prim( S ( R , q )) is a non-Hausdorfftriangle whose interior is Hausdorff, whose edges are doubled and whose verticesare triple points. Let us call the underlying Hausdorff quotient space T u /W D ,and its edges D α , D β and D γ , indicating their stabilizer.84 Chapter 6. Examples and calculations (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) D = D D bg Dv aa gb v v There is no need to indicate which of the points (6.34) is v α , as all configurationsoccur, for a suitable choice of a fundamental domain. C ∗ r ( W ) ∼ = (cid:8) f ∈ C (cid:0) D ; End( C [ W ]) (cid:1) : T s η f ( t ) T s η = f ( t ) ∀ t ∈ D η ,f ( v η ) ∈ End( C [ W ]) W ∀ η ∈ { α, β, γ } (cid:9) Consider the extension0 → C (cid:0) D, ∂D ; End( C [ W ]) (cid:1) → C ∗ r ( W ) → A → A = (cid:8) f ∈ C (cid:0) ∂D ; End( C [ W ]) (cid:1) : T s η f ( t ) T s η = f ( t ) ∀ t ∈ D η ,f ( v η ) ∈ End( C [ W ]) W ∀ η ∈ { α, β, γ } (cid:9) (6.35)In the vertices v η the A -representation C [ W ] is a direct sum of three irreducibles:the trivial W -representation, the sign representation of W and the defining (re-flection) representation of W (with multiplicity two). Likewise, on the edge D η C [ W ] is the direct sum of a part corresponding to the trivial representa-tion of { e, s η } and a part corresponding to the trivial representation of { e, s η } .Both summands have dimension three. Evaluating the reflection representationsat the vertices gives an extension0 → A → A → M ( C ) → A = { f ∈ C ( ∂D ; M ( C )) : f ( v η ) ∈ C ⊕ O } (6.36)where O is the 2 × A → C ( ∂D ; M ( C )) induces an isomorphism on K -theory, so we have an exact hexagon Z → K ( A ) → Z ↑ ↓ ← K ( A ) ← Z The vertical maps are 0, so K ( A ) ∼ = Z K ( A ) ∼ = Z .3. A Z → K (cid:0) C ∗ r ( W ) (cid:1) → Z ↑ ↓ Z ← K (cid:0) C ∗ r ( W ) (cid:1) ← K (cid:0) C ∗ r ( W ) (cid:1) ∼ = Z K (cid:0) C ∗ r ( W ) (cid:1) ∼ = Z (6.37)Define the projections p triv = (cid:80) w ∈ W T w p sign = (cid:80) w ∈ W ( − (cid:96) ( w ) T w We can unambiguously define classes of projections p , p , p by the requirements p i ( ζ j , ζ − j ) (cid:26) = p triv + p sign if i (cid:54) = j mod 3 ⊥ p triv + p sign if i = j mod 3Then K (cid:0) C ∗ r ( W ) (cid:1) is generated by[ p triv ] [ p sign ] [ p ] [ p ] [ p ]and a generator for K (cid:0) C ∗ r ( W ) (cid:1) is[ u ] = [ p triv N x p triv + p sign N − x p sign + T e − p triv − p sign ] • generic case q (cid:54) = • P = ∅ R P = ∅ R ∨ P = ∅ X P = X X P = 0 Y P = Y Y P = 0 T P = T T P = { } K P = { } W P = W ( P, P ) = W P P = W W P = { e } ı os η = ( T s η (1 − θ η ) + ( q − θ η )( q − θ η ) − η ∈ { α, β, γ } If s η ( t ) = t then ı os η = 1, and there are no points with stabilizer { e, s s , s s } , so C ∗ r ( R , q ) P ∼ = C ( D ; M ( C ))Prim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = T u /W ∼ = D • P = { α }
86 Chapter 6. Examples and calculations R P = {± α } R ∨ P = {± α ∨ } X P = X/ Z α ∼ = Z x X P = X/ Z x ∼ = Z α/ Y P = Z (0 , √ Y P = Z α ∨ T P = { ( t , t ) : t ∈ T } T P = { (1 , t ) : t ∈ T } K P = { (1 , , (1 , − } = { , k P } W p = { e, s α } W P = { e, s β , s α s β } W ( P, P ) = { e } W P P = K P The root datum R P is isomorphic to R ( A ) ∨ , so we can use the analysis on page174. The representations π ( P, δ (1 , t )) and π ( P, δ − , (1 , − t )) are intertwined by π ( k P ). C ∗ r ( R , q ) P ∼ = C ( S ; M ( C ))Prim( C ∗ r ( R , q ) P ) ∼ = (cid:0) P, W P ( q / , q / ) , δ , T Pu (cid:1) ∼ = S The corresponding central idempotent is p α + p β + p γ , where p η = (cid:80) w ∈ W η ( − q ) (cid:96) ( w ) T w (cid:16) (cid:80) w ∈ W η q ( w ) − (cid:17) − if q > p η = (cid:80) w ∈ W η T w (cid:16) (cid:80) w ∈ W η q ( w ) (cid:17) − if q < W η = (cid:104) s η , t η (cid:105) ⊂ W aff . • P = { β } This subset of F is conjugate to { α } by s α s β . • P = { α, β } R P = R R ∨ P = R ∨ X P = 0 X P = X Y P = 0 Y P = YT P = { } T P = T K P = { } W P = W ( P, P ) = W P P = { e } W P = W The residual points (6.33) all carry exactly one inequivalent discrete series repre-sentation. We have C ∗ r ( R , q ) ∼ = C with central idempotent p α,β = (cid:80) w ∈ W aff ( − q ) (cid:96) ( w ) T w (cid:16) (cid:80) w ∈ W aff q ( w ) − (cid:17) − if q > p α,β = (cid:80) w ∈ W aff T w (cid:16) (cid:80) w ∈ W aff q ( w ) (cid:17) − if q < p (1 , = ( T e + T ω + T ω ) p α,β / p ( ζ,ζ ) = ( T e + ζT ω + ζ T ω ) p α,β / p ( ζ ,ζ ) = ( T e + ζ T ω + ζT ω ) p α,β / A H ( W aff , q ).We conclude that the spectrum of C ∗ r ( R , q ) is the Hausdorff spacePrim (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = D (cid:116) S (cid:116) D is compact and contractible this implies K (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = Z (6.41)Let p and p be rank one projectors in C ∗ r ( R , q ) ∅ and C ∗ r ( R , q ) { α } . Then p , p ,p (1 , , p ( ζ,ζ ) and p ( ζ ,ζ ) generate K (cid:0) C ∗ r ( R , q ) (cid:1) , while T x generates K (cid:0) C ∗ r ( R , q ) (cid:1) .Once again, the K -theory turns out to be independent of the parameters. Interms of all the above generators, the group isomorphisms K ∗ ( φ ) are as follows. q < q = 1 q > p + p + p (1 , + p ( ζ,ζ ) + p ( ζ ,ζ ) p triv p p p sign p + p + p (1 , + p ( ζ,ζ ) + p ( ζ ,ζ ) p + p + p ( ζ,ζ ) + p ( ζ ,ζ ) p p + p + p ( ζ,ζ ) + p ( ζ ,ζ ) p + p + p (1 , + p ( ζ ,ζ ) p p + p + p (1 , + p ( ζ ,ζ ) p + p + p (1 , + p ( ζ,ζ ) p p + p + p (1 , + p ( ζ,ζ ) T x u T − x (6.42)As promised, we also discuss the root datum R ( A ), where X is the root lattice. X = Q = Z (1 ,
0) + Z (cid:0) / , √ / (cid:1) X + = { (cid:0) n + m/ , m √ / (cid:1) : 0 ≤ m ≤ n ≤ m } Y = Z (cid:0) , / √ (cid:1) + Z (cid:0) , / √ (cid:1) Q ∨ = Z (1 ,
0) + Z (cid:0) / , √ / (cid:1) R = {± α, ± β, ± γ } = {± (1 , , ± (cid:0) − / , √ / (cid:1) , ± (cid:0) / , √ / (cid:1) } = R R ∨ = (cid:8) ± α ∨ , ± β ∨ , ± γ ∨ (cid:9) = {± (1 , , ± (cid:0) − / , √ / (cid:1) , ± (cid:0) / , √ / (cid:1) } = R ∨ T = ( C × ) t = ( t , t ) = ( t ( α ) , t ( β )) F = { α, β } W = (cid:104) s α , s β | s α = s β = ( s α s β ) = e (cid:105) ∼ = S s = s α : ( n, m ) → ( − n, m ) s = s β : ( n + m/ , m √ / → ( m + n/ , n √ / s = t γ s γ : (cid:0) n + m/ , m √ / (cid:1) → (cid:0) (1 + n − m ) / , (1 − n − m ) √ / (cid:1) S aff = { s , s , s } W = W aff = (cid:104) s , W | s = ( s s ) = ( s s ) = e (cid:105) q ( s ) = q ( s ) = q ( s ) = q ( s ) = q α ∨ = q β ∨ = q γ ∨ := q
88 Chapter 6. Examples and calculationsGenerically there are 6 residual points and 6 tempered residual circles. Both forma single W -conjugacy class, typical examples being( q − , q − ) and { ( q − , t : t ∈ T } As usual we distinguish the cases q = 1 and q (cid:54) = 1. • group case q = The following subtori of T u have nontrivial stabilizers: { (1 , t ) : t ∈ T } { e, s α }{ ( t ,
1) : t ∈ T } { e, s β }{ ( t , t − ) : t ∈ T } { e, s γ } ( ζ, ζ ) , ( ζ , ζ ) { e, s α s β , s β s α } (1 , W The following part T (cid:48) of T u is a fundamental domain for the action of W . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) T’ = (1,1)(1,1) (z,z)
The left edge is T s α u , and to get T u /W we only have to identify the other twoedges by means of a rotation around ( ζ, ζ ). Then C ∗ r ( W ) consists of all f ∈ C (cid:0) T (cid:48) ; End( C [ W ]) (cid:1) such that1. T s α f ( t ) T s α = f ( t ) if t ∈ T s α u f (1 , ∈ (cid:0) End( C [ W ]) (cid:1) W f ( s β s α ( t , t )) = T s β s α f ( t t ) T s β s α ∀ t ∈ exp( πi [0 , / t ∈ T s α u then f ( t ) stabilizes C [ W ] s α , so there are extensions0 → A → C ∗ r ( W ) → A → → A → A → C → A = f ∈ C (cid:16) T s α u ; End (cid:0) C [ W ] s α (cid:1)(cid:17) : f (1 ,
1) = ∗ ∗ ∗ ∗ ∗ A = { f ∈ C ∗ r ( W ) : f (1 , ∈ C ⊕ O , f ( t ) ∈ M ( C ) ⊕ O ∀ t ∈ T s α u } A = { f ∈ A : f (1 , ∈ O ⊕ M ( C ) } .3. A O n denotes the n × n zero matrix. By Proposition 2.21 the inclusions A → A := { f ∈ C (cid:0) T (cid:48) ; End( C [ W ]) (cid:1) : 3. holds } A → C (cid:0) T s α u ; End (cid:0) C [ W ] s α (cid:1)(cid:1) (6.45)induce isomorphisms on K -theory. With the help of Lemma 2.26 we find that K ( A ) ∼ = Z K ( A ) = 0 K ( A ) ∼ = Z K ( A ) ∼ = Z K ( A ) ∼ = Z K ( A ) = 0 K ( A ) ∼ = Z K ( A ) ∼ = Z From (6.43) we get an exact hexagon Z → K (cid:0) C ∗ r ( W ) (cid:1) → Z ↑ ↓ Z ← K (cid:0) C ∗ r ( W ) (cid:1) ← K (cid:0) C ∗ r ( W ) (cid:1) ∼ = Z K (cid:0) C ∗ r ( W ) (cid:1) ∼ = Z (6.46)It is rather difficult to write down explicit generators, so we only indicate whatthey look like. Consider the projections p triv = (cid:80) w ∈ W T w p sign = (cid:80) w ∈ W ( − (cid:96) ( w ) T w p rot = ( T e + ζT s α s β + ζ T s β s α ) / p ζ ] and [ p ζ ] by the conditions p ζ (cid:0) T s α u (cid:1) = p ζ (cid:0) T s α u (cid:1) = p triv p ζ ( ζ, ζ )( p triv + p sign + p rot ) = 0 p ζ ( ζ, ζ ) p rot = p ζ ( ζ, ζ )These five classes of projections generate K (cid:0) C ∗ r ( W ) (cid:1) , and a generator for K (cid:0) C ∗ r ( W ) (cid:1) is u = p triv θ β p triv + p sign θ − β p sign + T e − p triv − p sign • generic case q (cid:54) = • P = ∅
90 Chapter 6. Examples and calculations R P = ∅ R ∨ P = ∅ X P = X X P = 0 Y P = Y Y P = 0 T P = T T P = { } K P = { } W P = W ( P, P ) = W P P = W W P = { e } ı os η = ( T s η (1 − θ η ) + ( q − θ η )( q − θ η ) − η ∈ { α, β, γ } If s η ( t ) = t then ı os η ( t ) = 1. There are two points in T u whose stabilizer is notgenerated by reflections: ( ζ, ζ ) and ( ζ , ζ ). Let A be as in (6.45). Then C ∗ r ( R , q ) P ∼ = A Prim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) which is supposed to depict T u /W with in the center, instead of just W ( ζ, ζ ), atriple non-Hausdorff point. By Lemma 2.26 we have K (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) P (cid:1) = 0The class of a projection in this algebra is completely determined by its value at( ζ, ζ ), and over there we only have to say which representation of { e, s s , s s } it gives. So we have generators p , p , p with p i ( ζ, ζ ) ı os s ( ζ, ζ ) = ζ i p i ( ζ, ζ ) • P = { α } R P = {± α } R ∨ P = {± α ∨ } X P = X/ Z α ∼ = Z β X P = X/ Z (0 , ∼ = Z ( α/ Y P = Z (0 , / √ Y P = Z α ∨ = Z (2 , T P = { (1 , t ) : t ∈ C × } T P = { ( t − , t ) : t ∈ C × } K P = { (1 , , (1 , − } = { , k P } W P = { e, s α } W P = { e, s β , s α s β } W ( P, P ) = { e } W P P = K P The root datum R P is isomorphic to R ( A ) ∨ , so we already know its discreteseries. The representations π ( P, δ , (1 , t )) and π ( P, δ − , (1 , − t )) are interwinedby π ( k P ), so Prim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = (cid:0) P, W P ( q, q / ) , δ , T Pu (cid:1) ∼ = S C ∗ r ( R , q ) P ∼ = C ( S ; M ( C ))The central idempotent for this component is p α + p β + p γ , as given by (6.39)..4. B • P = { β } This subset of F is conjugate to { α } by s α s β . • P = { α, β } R P = R R ∨ P = R ∨ X P = 0 X P = XY P = 0 Y P = YT P = { } T P = T K P = { } W P = W W P = W ( P, P ) = W P P = { e } Up to W -conjugacy there is a single residual point, so by Proposition 3.20.2 thereis a unique discrete series representation δ . We have C ∗ r ( R , q ) P ∼ = C , and thecorresponding projector is p δ = (cid:80) w ∈ W ( − q ) (cid:96) ( w ) T w (cid:16) (cid:80) w ∈ W q ( w ) − (cid:17) − if q > p δ = (cid:80) w ∈ W T w (cid:16) (cid:80) w ∈ W q ( w ) (cid:17) − if q < K (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = Z (6.48)which is the same as for q = 1. Generators are the invertible θ β and the rank oneprojectors p , p , p , p δ and p { α } .The isomorphisms K ∗ ( φ ) take the following form: q < q = 1 q > p + p { α } + p δ p triv p p p sign p + p { α } + p δ p + p { α } p rot p + p { α } p + p { α } + p δ p ζ p p + p { α } + p δ p ζ p θ β u θ − β (6.49) B B is probably the best testcase for Conjecture 5.28.Because there are roots of different lengths the conjecture is not yet known, andat the same time the calculations are still manageable. Moreover some interestingphenomena already occur for these affine Hecke algebras, like residual points thatcarry several inequivalent discrete series representations.92 Chapter 6. Examples and calculationsWe will only consider the root datum R ( B ) ∨ where X is the weight lattice: X = Z Q = { ( m, n ) ∈ Z : n + m is even } X + = { ( m, n ) ∈ Z : n ≥ m ≥ } Y = Q ∨ = Z T = ( C × ) t = ( t , t ) = ( t (1 , , t (0 , R = R = {± α , ± α , ± α , ± α } = {± (2 , , ± ( − , , ± (1 , , ± (0 , } R ∨ = R ∨ = {± α ∨ , ± α ∨ , ± α ∨ , ± α ∨ } = {± (1 , , ± ( − , , ± (1 , , ± (0 , } F = { α , α } W = (cid:104) s , s | s = s = ( s s ) = e (cid:105) ∼ = D s i = s α i s = t α s α = ( t (1 , s s ) s ( t (1 , s s ) − : ( m, n ) → (1 − n, − m ) S aff = { s , s , s } W (cid:54) = W aff = (cid:104) s , W | s = ( s s ) = ( s s ) = e (cid:105) Ω = { e, t (1 , s } = { e, ω } q := q ( s ) = q α ∨ = q α ∨ q := q ( s ) = q ( s ) = q α ∨ = q α ∨ c α i = (1 − q − α ∨ i θ − α i )(1 − θ − α i ) − i = 1 , , , (cid:8) ( q / , t ) : t ∈ T (cid:9) (6.50) (cid:8) ( − q / , t ) : t ∈ T (cid:9) (6.51) (cid:8) ( q / t , q − / t ) : t ∈ T (cid:9) (6.52)( q − / , q / q − ) , ( q / , q − / q − ) , ( − q − / , − q / q − ) , ( − q / , − q − / q − ) , ( − q − / , q / ) (6.53)It turns out that there are five classes of parameters with the same level of gener-icity. The first three are easily found from (6.50) - (6.52): q = 1 = q , q (cid:54) = 1 = q and q = 1 (cid:54) = q . Furthermore we have the generic class and the four special lines q = q (cid:54) = 1 , q = q − (cid:54) = 1 , q = q (cid:54) = 1 , q = q − (cid:54) = 1 (6.54) • group case q = q = From (6.2) and (6.3) we see that it pays off to determine the extended quotient (cid:102) T u /W ∼ = (cid:71) (cid:104) w (cid:105)∈(cid:104) W (cid:105) T wu (cid:14) Z W ( w )There are five conjugacy classes, namely { e } , { s , s } , { s , s } , { s s , s s } , { s s s s } .4. B w Z W ( w ) T wu T wu /Z W ( w ) e W T u { ( t , t ) ∈ [0 , : t ≥ t } s { e, s , s , s s } { ( ± , t ) ∈ T u } {± } × [ − , s { e, s , s , s s } { ( t , t ) ∈ T u } [ − , s s (cid:104) s s (cid:105) { (1 , , ( − , − } { (1 , , ( − , − } ( s s ) W { ( ± , ± , ( ± , ∓ } { (1 , , ( − , − , (1 , − } Since all components of this space are contractible, with (6.3) we find that K (cid:0) C ∗ r ( W ) (cid:1) ∼ = ˇ H ∗ (cid:0)(cid:102) T u /W ; Z (cid:1) ∼ = Z K (cid:0) C ∗ r ( W ) (cid:1) = 0 (6.55)We may visualize the extended quotient as (cid:102) T u /W ∼ = (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) In the same way we havePrim (cid:0) C ∗ r ( W ) (cid:1) ∼ = (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (-1,1)(1,1) (-1,-1) where the multiplicities of the non-Hausdorff points at the edge can be read offfrom the previous picture, by collapsing everything on the triangle.To find generating projections for the K -theory we first have a closer lookat W . This group has four one-dimensional representations, let us call them (cid:15) i .94 Chapter 6. Examples and calculationsThese representations and the corresponding projections are (cid:15) i (cid:15) i ( s ) (cid:15) ( s ) p i (cid:15) ( T e + T s s + T s s + T ( s s ) + T s + T s + T s + T s ) (cid:15) − ( T e + T s + T s + T ( s s ) − T s − T s − T s s − T s s ) (cid:15) − ( T e + T s + T s + T ( s s ) − T s − T s − T s s − T s s ) (cid:15) − − ( T e + T s s + T s s + T ( s s ) − T s − T s − T s − T s )(6.56)The remaining irreducible W -representation ρ has dimension two, it defines W as a reflection group. Let p ∈ C [ W ] be rank one projector for that representation,i.e. ρ ( p ) = (cid:18) (cid:19) and e i ( p ) = 0 i = 0 , , , p has rank two in End C (cid:0) C [ W ] (cid:1) .We also have to consider the stabilizer of ( − , ∈ T , the subgroup { e, s , s , ( s s ) } ∼ = D . Let us list its irreducible representations, which all havedimension one: (cid:15) (cid:15) ( s ) (cid:15) ( s ) Ind W (cid:104) s ,s (cid:105) ( (cid:15) ) p ( (cid:15) ) (cid:15) ++ (cid:15) ⊕ (cid:15) p (cid:15) + − − ρ p + − (cid:15) − + − ρ p − + (cid:15) −− − − (cid:15) ⊕ (cid:15) p (6.57)In the last column we indicate an element of End C (cid:0) C [ W ] (cid:1) that acts as a rankone projector for that representation of (cid:104) s , s (cid:105) . For (cid:15) + − and (cid:15) − + such an elementcannot be found in C [ W ] ∼ = End W (cid:0) C [ W ] (cid:1) , so we refrain from giving an explicitformula.Now we can indicate generators p , . . . , p for K (cid:0) C ∗ r ( W ) (cid:1) . The last four classesof projections are defined by their values in three special points. p p (1 , p ( − , − p ( − , p p p p − + p p p p + − p p + p p p + p p p + p p p + − + p − + • generic case • P = ∅ R P = ∅ R ∨ P = ∅ X P = X X P = 0 Y P = Y Y P = 0 T P = T T P = { } K P = { } W P = W ( P, P ) = W P P = W W P = { e } ı os i = ( T s i (1 − θ α i ) + ( q ( s i ) − θ α i )( q ( s i ) − θ α i ) − i = 1 , , , ı os i ( t ) = 1 if t ∈ T s i .4. B W -stabilizer of any t ∈ T is generated by reflections, so all the unitaryrepresentations π ( ∅ , δ ∅ , t ) are irreducible. C ∗ r ( R , q ) P ∼ = C ( T u /W ; M ( C ))Prim (cid:0) C ∗ r ( R , q ) P (cid:1) = T u /W ∼ = (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (-1,1)(1,1) (-1,-1) Since T u /W is contractible we conclude that K (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) P (cid:1) = 0A generator is p ∅ = e ∅ (cid:88) w ∈ W q ( w ) T w • P = { α } R P = {± α } R ∨ P = {± α ∨ } X P = X/ Z ( α / ∼ = Z α / X P = X/ Z ( α / ∼ = Z α / Y P = Z α ∨ Y P = Z α ∨ T P = { (1 , t ) : t ∈ C × } T P = { ( t ,
1) : t ∈ C × } K P = { } W P = { e, s } W P = { e, s , s , s s } W ( P, P ) = W P P = { e, s } ı os = ( T s (1 − θ α ) + ( q − θ α )( q − θ α ) − The algebra H P is isomorphic to H ( R ( A ) ∨ , q ). Hence it has two discrete seriesrepresentations, with central characters (cid:0) q ± / , (cid:1) and (cid:0) − q ± / , (cid:1) . Clearly ı os ( t ) = 1 if t ∈ (cid:0) T Pu (cid:1) s , so C ∗ r ( R , q ) P ∼ = C ([0 , M ( C )) Prim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = K (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) P (cid:1) = 096 Chapter 6. Examples and calculationsLet us denote the canonical generators by [ p + α ] and [ p − α ]. Notice that as H ( W , q )-representations the π ( P, δ, t ) are deformations of Ind W W P (sign W P ) if q > W W P (triv W P ) if q < • P = { α } R P = {± α } R ∨ P = {± α ∨ } X P = X/ Z α ∼ = Z α / X P = X/ Z α ∼ = Z α / Y P = Z α ∨ Y P = Z α ∨ T P = { ( t , t ) : t ∈ C × } T P = { ( t − , t ) : t ∈ C × } K P = { (1 , , ( − , − } = { , k P } W P = { e, s } W P = { e, s , s , s s } W ( P, P ) = { e, s } ı os = ( T s (1 − θ α ) + ( q − θ α )( q − θ α ) − ı os ( t ) = 1 if t ∈ (cid:0) T Pu (cid:1) s The algebra H P is isomorphic to H ( R ( A ) ∨ , q ), so it has two discrete seriesrepresentations δ + and δ − . Their central characters are respectively ( q ± / , q ∓ / )and ( − q ± / , − q ∓ / ). However π ( k P ) intertwines π ( P, δ + , ( t , t )) and π ( P, δ − , ( − t , − t )), so in the spectrum we get only one component ( P, δ + , T Pu ).The intertwiner π ( s ) acts as a reflection on T Pu and π ( s , P, δ + , t ) = 1 whenever s ( t ) = t . Therefore C ∗ r ( R , q ) P ∼ = C ([0 , M ( C ))Prim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = ( P, δ + , T Pu ) /W ( P, P ) ∼ = [0 , K (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) P (cid:1) = 0There is a canonical generator [ p α ] ∈ K (cid:0) C ∗ r ( R , q ) P (cid:1) . The type of π ( P, δ + , t ) asa representation of H ( W , q ) is easily determined: for q < W W P (triv W P ) while for q > W W P (sign W P ). • P = { α , α } R P = R R ∨ P = R ∨ X P = 0 X P = X Y P = 0 Y P = YT P = { } T P = T K P = { } W P = W ( P, P ) = W P P = { e } W P = W Generically all the residual points (6.53) are in orbits consisting of | W | = 8points. Proposition 3.20.1 tells us that every such W -orbit is the central characterof precisely one discrete series representation. The discrete series representation.4. B δ with central character W ( q − / , − q − / ) is the easiest to describe. It hasdimension two, and as a W -representation δ ◦ φ is equivalent to (cid:15) ⊕ (cid:15) (for q > (cid:15) ⊕ (cid:15) (for q < δ , δ , δ , δ be the discrete seriesrepresentations with respective central characters W (cid:0) q / , q − / q (cid:1) , W (cid:0) q / , q / q (cid:1) , W (cid:0) − q / , − q − / q (cid:1) , W (cid:0) − q / , − q / q (cid:1) We list the type of δ i ◦ φ as a representation of W ⊂ S ( W ), for different q ’s : δ δ δ δ < q / < q < q ρ (cid:15) ρ (cid:15) q − < q < q − / < (cid:15) ρ (cid:15) ρ < q − / < q < q − (cid:15) ρ (cid:15) ρ q < q < q / < ρ (cid:15) ρ (cid:15) q − / < q < q / > (cid:15) (cid:15) (cid:15) (cid:15) q − < q < q > (cid:15) (cid:15) (cid:15) (cid:15) > q / < q < q − / (cid:15) (cid:15) (cid:15) (cid:15) > q < q < q − (cid:15) (cid:15) (cid:15) (cid:15) (6.58)It can be shown by direct calculation that this table still gives all discrete seriesrepresentations if either q or q (but not both) equals 1. This not true for thespecial parameters (6.54) however. Nevertheless, for all the parameters underconsideration here K (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) P (cid:1) = 0We denote the generating projections by p ( δ i ) , i = 1 , , , , C ∗ r ( R , q ) is Morita-equivalent to the commutative C ∗ -algebra with spectrumPrim (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) and that K (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) (cid:1) = 0 (6.59)98 Chapter 6. Examples and calculations • q (cid:54) = = q • P = ∅ R P = ∅ R ∨ P = ∅ X P = X X P = 0 Y P = Y Y P = 0 T P = T T P = { } K P = { } W P = W ( P, P ) = W P P = W W P = { e } ı os = ( T s (1 − θ α ) + ( q − θ α )( q − θ α ) − ı os = T s Since ı os ( t ) = 1 if s ( t ) = t , all the nonscalar selfintertwiners of unitary principalseries representations come from s or its conjugates. We see that π ( ∅ , δ ∅ , t ) isirreducible unless t ∈ T s u ∪ T s u , in which case it is the direct sum of two inequivalentsubrepresentations. Hence C ∗ r ( R , q ) P ∼ = (cid:8) f ∈ C (cid:0) T u /W ; M ( C ) (cid:1) : f (cid:0) T s u (cid:1) ∈ M ( C ) ⊕ M ( C ) (cid:9) Prim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) In this picture the diagonal edge should be regarded as consisting of double points.The algebra is diffeotopy equivalent to M ( C ) , so K (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) P (cid:1) = 0Generators are for example p := 18 e ∅ (cid:88) w ∈ W q ( w ) T w and p := 18 e ∅ (cid:88) w ∈ W ( − (cid:96) ( w ) T w • P = { α } This is identical to P = { α } in the generic case. • P = { α } Here H P ∼ = H ( R ( A ) ∨ , q ) = C [ W ( A )]. As we saw before, this algebra has nodiscrete series representations, so there is no component in the spectrum of S ( R , q )corresponding to this P ..4. B • P = { α , α } R P = R R ∨ P = R ∨ X P = 0 X P = X Y P = 0 Y P = YT P = { } T P = T K P = { } W P = W ( P, P ) = W P P = { e } W P = W Some residual points confluence when q →
1, and only three orbits remain. Twoof those consist of four points, represented by (cid:0) q − / , q / (cid:1) and (cid:0) − q − / , − q − / (cid:1) .The last orbit still contains 8 different points, for example (cid:0) − q − / , q / (cid:1) . ByProposition 3.20.1 there is exactly one discrete series representation δ with centralcharacter W (cid:0) − q − / , q / (cid:1) . It has dimension two and restricted to H ( W , q ) it isa deformation of the W -representations (cid:15) ⊕ (cid:15) or (cid:15) ⊕ (cid:15) , depending on whether q > q <
1. We calculated the other discrete series representations alreadyin (6.58). For q > δ δ ( T s ) δ ( T s ) δ ( θ x ) δ a − q − / , q − / )( x ) δ b − − q − / , q − / )( x ) δ c − − q − / , − q − / )( x ) δ d − − − q − / , − q − / )( x ) (6.60)On the other hand, for q < δ δ ( T s ) δ ( T s ) δ ( θ x ) δ a q q / , q / )( x ) δ b q − q / , q / )( x ) δ c q − q / , − q / )( x ) δ d q − − q / , − q / )( x ) (6.61)This leads to C ∗ r ( R , q ) P ∼ = M ( C ) ⊕ C K (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) P (cid:1) = 0The generators of K (cid:0) C ∗ r ( R , q ) P (cid:1) corresponding to rank one projectors in theserepresentations are denoted by p ( δ v ) , v ∈ { a, b, c, d, } .00 Chapter 6. Examples and calculationsCombining all P ’s we find K (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) P (cid:1) = 0Prim (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) • q = (cid:54) = q • P = ∅ R P = ∅ R ∨ P = ∅ X P = X X P = 0 Y P = Y Y P = 0 T P = T T P = { } K P = { } W P = W ( P, P ) = W P P = W W P = { e } ı os = T s ı os = ( T s (1 − θ α ) + ( q − θ α )( q − θ α ) − Since ı os ( t ) = 1 whenever t ∈ T s , all the nonscalar self-intertwiners of principalseries representations come from s , s = s s s and s s . This implies that weshould divide the points t ∈ T u in five classes.1. s ( t ) (cid:54) = t (cid:54) = s ( t )Here we do not encounter nonscalar self-intertwiners, so π ( ∅ , δ ∅ , t ) is irre-ducible.2. s ( t ) = t (cid:54) = s ( t )For such t π ( ∅ , δ ∅ , t ) splits into two summands, which as H ( W , q )-representationsare deformations of Ind W { e,s } (triv) and of Ind W { e,s } (sign).3. s ( t ) (cid:54) = t = s ( t )Just as in 2. π ( ∅ , δ ∅ , t ) is the direct sum of two irreducible subrepresen-tations, which can be regarded as deformations of Ind W { e,s } (triv) and ofInd W { e,s } (sign).4. (1 ,
1) and ( − , − W -invariant, so s and s are conjugate in W ,t . Hence π ( s , ∅ , δ ∅ , t ) is essentially the only independent intertwiner, and π ( ∅ , δ ∅ , t ).4. B W -representations (cid:15) ⊕ (cid:15) ⊕ ρ and (cid:15) ⊕ (cid:15) ⊕ ρ .5. (1 , −
1) and ( − , W -orbit in T u , their stabilizer being { e, s , s , s s } .Here the intertwiners π ( e ) , π ( s ) , π ( s ) , π ( s s ) are all linearly independent,so π ( ∅ , δ ∅ , t ) splits into no less than four irreducible summands, correspond-ing to the irreducible representations of { e, s , s , s s } ∼ = ( Z / Z ) .We visualize this asPrim (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) where the lower right corner depicts the fourfold non-Hausdorff point ( − , − , K (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) P (cid:1) = 0Generators are the rank one projectors p = 18 (cid:88) w ∈ W q ( w ) T w , p = 18 (cid:88) w ∈ W ( − (cid:96) ( w ) T w , p − + , p + − the last two being defined in the same way as the homonymous (classes of) pro-jections in (6.57). • P = { α } Here H P ∼ = H ( R ( A ) ∨ , q ) ∼ = C [ W ( A )], so we do not find any discrete seriesrepresentations to induce. • P = { α } This is identical to P = { α } for generic labels. • P = { α , α } R P = R R ∨ P = R ∨ X P = 0 X P = X Y P = 0 Y P = YT P = { } T P = T K P = { } W P = W ( P, P ) = W P P = { e } W P = W
02 Chapter 6. Examples and calculationsIn the limit q → W -orbits of four points, from which we pick the repre-sentatives (1 , q ) and ( − , − q ). The other discrete series representations werealready constructed in (6.58). They have dimension one, and for q > δ δ ( T s ) δ ( T s ) δ ( θ x ) δ a − , q − )( x ) δ b − − , q − )( x ) δ c − − , − q − )( x ) δ d − − − , − q − )( x ) (6.62)On the other hand, for q < δ δ ( T s ) δ ( T s ) δ ( θ x ) δ a q (1 , q − )( x ) δ b − q (1 , q − )( x ) δ c q ( − , − q − )( x ) δ d − q ( − , − q − )( x ) (6.63)This summand of C ∗ r ( R , q ) is actually commutative: C ∗ r ( R , q ) P ∼ = C K (cid:0) C ∗ r ( R , q ) P (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) P (cid:1) = 0Thus the spectrum of C ∗ r ( R , q ) is the non-Hausdorff spacePrim (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) and its K -groups are K (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = Z K (cid:0) C ∗ r ( R , q ) (cid:1) = 0 (6.64).4. B • q = q (cid:54) = , q = q − (cid:54) = , q = q (cid:54) = , q = q − (cid:54) = These parameters are a bit too tricky to analyse with techniques we used so far.To determine the spectra of the corresponding C ∗ -algebras one can use the preciseresults on R-groups in [122]. It turns out that there are only three inequivalentdiscrete series representations. On the other hand, compared to the generic casetwo of the representations π ( P, δ, t ) with | P | = 1 become reducible.Let us make up the balance for the root datum R ( B ) ∨ . The K -groups vanishfor all label functions, and the K -groups are all free abelian of rank 9. We did notgive all the generating projections explicitly, but we have enough information todetermine the maps K ( φ ). In the next table we list the images of the generators p i of K ( S ( W )). Assuming that all the calculations in this section are correct, thetable shows that Conjecture 5.28 is valid for R ( B ) ∨ .We will not discuss the root datum R ( B ) exhaustively, because there too aremany different label functions. The group case is as in this section, and the equallabel case is well understood, as described in Section 5.1. We will study the genericlabel case in Section 6.7.04 Chapter 6. Examples and calculations < q − / < q < q − q = 1 < q < q / < q < q p ∅ + p + α + p − α + p ( δ ) p p ∅ p ∅ p p ∅ + p + α + p − α + p ( δ ) p ∅ + p + α + p − α + p α + p ( δ ) + p ( δ ) + p ( δ ) p + p α + p ( δ a ) + p ( δ c ) p ∅ + p α p ∅ + p α p + p α + p ( δ b ) + p ( δ d ) p ∅ + p + α + p − α + p α + p ( δ ) + p ( δ ) + p ( δ )2 p ∅ + p + α + p − α + p ( δ ) + p ( δ ) p − + + p + − + p α p ∅ + p + α + p − α + p ( δ ) + p ( δ ) p ∅ + p + α p − + p ∅ + p − α p ∅ + p + α + p α + p ( δ ) p + − + p α + p ( δ a ) + p ( δ d ) p ∅ + p − α + p α + p ( δ )2 p ∅ + p + α + p − α + p α + p ( δ ) + p ( δ ) p + p + p α + p ( δ b ) 2 p ∅ + p + α + p − α + p α + p δ + p ( δ ) + p ( δ )2 p ∅ + p + α + p − α + p α + p ( δ ) p + − + p − + + p α + p ( δ b ) 2 p ∅ + p + α + p − α + p α + p δ + p ( δ ) q < q q = 1 = q q > q p + p + α + p − α + p ( δ a ) + p ( δ c ) + p ( δ ) p p p p p + p + α + p − α + p ( δ a ) + p ( δ c ) + p ( δ ) p + p + α + p − α + p ( δ b ) + p ( δ d ) + p ( δ ) p p p p p + p + α + p − α + p ( δ b ) + p ( δ d ) + p ( δ ) p + p + p + α + p − α p p + p + p + α + p − α p + p + α + p ( δ a ) p p + p − α + p ( δ c ) p + p + α + p ( δ b ) p p + p − α + p ( δ d ) p + p + p + α + p − α + p ( δ a ) + p ( δ ) p p + p + p + α + p − α + p ( δ b ) + p ( δ ) p + p + p + α + p − α + p ( δ a ) p p + p + p + α + p − α + p ( δ b ) q < q < q / < q = 1 > q q − < q < q − / < p ∅ + p + α + p − α + p α + p ( δ ) + p ( δ ) + p ( δ ) p + p α + p ( δ a ) + p ( δ c ) p ∅ + p α p ∅ + p α p + p α + p ( δ b ) + p ( δ d ) p ∅ + p + α + p − α + p α + p ( δ ) + p ( δ ) + p ( δ ) p ∅ + p + α + p − α + p ( δ ) p p ∅ p ∅ p p ∅ + p + α + p − α + p ( δ )2 p ∅ + p + α + p − α + p α + p ( δ ) + p ( δ ) p + − + p − + p ∅ + p + α + p − α + p α + p ( δ ) + p ( δ ) p ∅ + p + α + p α + p ( δ ) p − + + p α + p ( δ a ) + p ( δ d ) p ∅ + p − α + p α + p ( δ ) p ∅ + p + α p + − p ∅ + p − α p ∅ + p + α + p − α + p α + p ( δ ) + p ( δ ) + p ( δ ) p + p + p α + p ( δ a ) 2 p ∅ + p + α + p − α + p α + p ( δ ) + p ( δ )2 p ∅ + p + α + p − α + p α + p ( δ ) + p ( δ ) p + − + p − + + p α + p ( δ a ) 2 p ∅ + p + α + p − α + p α + p ( δ ) .5. GL n GL n After the calculations with twodimensional root data we move on to higher ranks.The easiest root data to study are those associated with the reductive group GL n .The way right way to do this was indicated in [17]. From [103, Lemma 5.3] weknow that the topological K -groups of these affine Hecke algebras are free abelian,and according to Theorem 5.3 they do not depend on q . Because of the higherdimensionality we do not provide explicit generators for these K -groups anymore.Nevertheless, by a different argument we show that Conjecture 5.28 holds for theseroot data.From now on many things will be parametrized by partitions and permutations,so let us agree on some notations. We write partitions in decreasing order andabbreviate ( x ) = ( x, x, x ). A typical partition looks like µ = ( µ , µ , . . . , µ d ) = ( n ) m n · · · (2) m (1) m (6.65)where some of the multiplicities m i may be 0. By µ (cid:96) n we mean that the weightof µ is | µ | = µ + · · · + µ d = n The number of different µ i ’s (i.e. the number of blocks in the diagram of µ ) willbe denoted by b ( µ ) and the dual partition (obtained by reflecting the diagram of µ ) by µ ∨ . Sometimes we abbreviate g = gcd( µ ) = gcd( µ , . . . , µ d ) µ ! = µ ! µ ! · · · µ d ! (6.66)With a such partition µ of n we associate the permutation σ ( µ ) = (12 · · · µ )( µ + 1 · · · µ + µ ) · · · ( n + 1 − µ d · · · n ) ∈ S n As is well known, this gives a bijection between partitions of n and conjugacyclasses in the symmetric group S n . The centralizer Z S n ( σ ( µ )) is generated by thecycles (( µ + · · · + µ i + 1)( µ + · · · + µ i + 2) · · · ( µ + · · · + µ i + µ i +1 ))and the “permutations of cycles of equal length”, for example if µ = µ :(1 µ + 1)(2 µ + 2) · · · ( µ µ ) (6.67)Using the second presentation of µ this means that Z S n ( σ ( µ )) ∼ = n (cid:89) l =1 ( Z /l Z ) m l (cid:111) S m l
06 Chapter 6. Examples and calculationsLet us recall the definition of R ( GL n ) : X = Z n Q = { x ∈ X : x + · · · + x n = 0 } X + = { x ∈ Z n : x ≥ x ≥ · · · ≥ x n } Y = Z n Q ∨ = { y ∈ Y : y + · · · + y n = 0 } T = ( C × ) n t = ( t ( e ) , . . . , t ( e n )) = ( t , . . . , t n ) R = R = { e i − e j ∈ X : i (cid:54) = j } R ∨ = R ∨ = { e i − e j ∈ Y : i (cid:54) = j } F aff = { α ∨ i = e i − e i +1 } ∪ { − α ∨ = 1 − ( e − e n ) } s i = s α i s = t α s α = t − α s α t α : x → x + α − (cid:104) α ∨ , x (cid:105) α W = (cid:104) s , · · · , s n − | s i = ( s i s i +1 ) = ( s i s j ) = e : | i − j | > (cid:105) ∼ = S n S aff = { s , s , . . . s n − } W aff = (cid:104) s , W | s = ( s s i ) = ( s s ) = ( s s n − ) = e if 2 ≤ i ≤ n − (cid:105) W = W aff (cid:111) Ω Ω = (cid:104) t e (1 2 · · · n ) (cid:105) ∼ = Z Because all roots of R are conjugate, s is conjugate to any s i ∈ S aff . Hence forany label function we have q ( s ) = q ( s i ) = q α ∨ i := q The W -stabilizer of a point (cid:0) ( t ) µ ( t µ +1 ) µ · · · ( t n ) µ d (cid:1) ∈ T is isomorphic to S µ × S µ × · · · × S µ d . • group case q = By (6.2) we have K ∗ (cid:0) C ∗ r ( W ) (cid:1) ⊗ C ∼ = ˇ H ∗ (cid:0)(cid:102) T u (cid:14) S n ; C (cid:1) ∼ = (cid:77) µ (cid:96) n ˇ H ∗ (cid:0) T σ ( µ ) u (cid:14) Z S n ( σ ( µ )); C (cid:1) Therefore we want to determine T σ ( µ ) u /Z S n ( σ ( µ )). If µ is as in (6.65) then T σ ( µ ) = { ( t ) µ ( t µ +1 ) µ · · · ( t n ) µ d ∈ T } T σ ( µ ) (cid:14) Z S n ( σ ( µ )) ∼ = ( C × ) m n /S m n × · · · × ( C × ) m /S m (6.68)where S m l acts on ( C × ) m l by permuting the coordinates. To handle this spacewe use the following nice, elementary result, a proof of which can be found forexample in [17, p. 97]. Lemma 6.1
For any m ∈ N there is an isomorphism of algebraic varieties ( C × ) m (cid:14) S m ∼ = C m − × C × .5. GL n T σ ( µ ) u (cid:14) Z S n ( σ ( µ )) has the homotopy type of T b ( µ ) . The latterspace has torsion free cohomology, so by (6.3) K ∗ (cid:0) C ∗ r ( W ) (cid:1) ∼ = ˇ H ∗ (cid:0)(cid:102) T u /S n ; Z (cid:1) ∼ = (cid:77) µ (cid:96) n ˇ H ∗ (cid:0) T b ( µ ) ; Z (cid:1) ∼ = (cid:77) µ (cid:96) n Z b ( µ ) (6.69) • generic, equal label case q (cid:54) = 1 • P µ = F \ { α µ , α µ + µ , . . . , α n − µ d } Inequivalent subsets of F are parametrized by partitions µ of n . For the typicalpartition (6.65) we put R P µ ∼ = ( A n − ) m n × · · · × ( A ) m ∼ = R ∨ P µ X P µ ∼ = Z ( e + · · · + e µ ) /µ + · · · + Z ( e n +1 − µ d + · · · + e n ) /µ d X P µ ∼ = (cid:0) Z n / Z ( e + · · · + e n ) (cid:1) m n × · · · × (cid:0) Z / Z ( e + e ) (cid:1) m Y P µ = Z ( e + · · · + e µ ) + · · · + Z ( e n +1 − µ d + · · · + e n ) Y P µ = { y ∈ Z n : y + · · · + y µ = · · · = y n +1 − µ d + · · · + y n = 0 } T P µ = { ( t ) µ · · · ( t n ) µ d ∈ T } T P µ = { t ∈ T : t t · · · t µ = · · · = t n +1 − µ d · · · t n = 1 } K P µ = { t ∈ T P µ : t µ = · · · = t µ d n = 1 } W P µ ∼ = ( S n ) m n × · · · × ( S ) m W ( P µ , P µ ) ∼ = S m n × · · · × S m × S m W P µ P µ = K P µ (cid:111) W ( P µ , P µ ) Z S n ( σ ( µ )) = W ( P µ , P µ ) (cid:110) n (cid:89) l =1 ( Z /l Z ) m l The W P µ -orbits of residual points for H P µ are parametrized by K P µ (cid:0) ( q ( µ − / , q ( µ − / , . . . , q (1 − µ ) / ) · · · ( q ( µ d − / , q ( µ d − / , . . . , q (1 − µ d ) / ) (cid:1) This set is obviously in bijection with K P µ , and indeed the intertwiners π ( k ) , k ∈ K P µ act on it by multiplication. Together with Proposition 3.20.2 this implies (cid:91) ∆ Pµ (cid:0) P µ , δ, T P µ (cid:1)(cid:14) K P µ ∼ = T P µ (cid:91) ∆ Pµ (cid:0) P µ , δ, T P µ (cid:1)(cid:14) W P µ P µ ∼ = T P µ (cid:14) W ( P µ , P µ ) = T σ ( µ ) (cid:14) Z S n ( σ ( µ ))It a point t ∈ T P µ has a nontrivial stabilizer in W ( P µ P µ ) ∼ = (cid:81) nl =1 S m l , thenthis isotropy group is generated by transpositions. From (6.67) we see that everysuch transposition w ∈ W can be written as a product of mutually commutingreflections s α with c − α ( t ) = 0. By (3.122) this gives ı ow ( t ) = 1, and since dim δ = 1also π ( w, P µ , δ, t ) = 1 if w ( t ) = t (6.70)08 Chapter 6. Examples and calculationsSo the action of W P µ P µ on C (cid:16) (cid:71) ∆ Pµ T P µ u ; M n ! /µ ! ( C ) (cid:17) is essentially only on (cid:70) ∆ Pµ T P µ u and the conjugation part doesn’t really matter.In particular we deduce that C ∗ r ( R , q ) ∼ = (cid:77) µ (cid:96) n M n ! /µ ! (cid:16) C (cid:16) (cid:71) ∆ Pµ T P µ u (cid:17)(cid:17) ∼ = (cid:77) µ (cid:96) n M n ! /µ ! (cid:0) T σ ( µ ) u (cid:14) Z S n ( σ ( µ )) (cid:1) (6.71)Similar results were obtained by completely different methods in [93]. Just as inthe group case it follows that K ∗ (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = (cid:77) µ (cid:96) n K ∗ (cid:0) T σ ( µ ) u (cid:14) Z S n ( σ ( µ )) (cid:1) ∼ = (cid:77) µ (cid:96) n K ∗ (cid:0) T b ( µ ) (cid:1) ∼ = (cid:77) µ (cid:96) n Z b ( µ ) (6.72)As promised, we show that Conjecture 5.28 holds in this particular case. Theorem 6.2 K ∗ ( φ ) : K ∗ (cid:0) C ∗ r ( R ( GL n ) , q ) (cid:1) → K ∗ (cid:0) C ∗ r ( R ( GL n ) , q ) (cid:1) is an isomorphism.Proof. We assume that q >
1. If instead q < J ci ⊂ C ∗ r ( R , q ) be the norm completion of the ideal J i ⊂ S ( R , q ) from(3.144). For every i there is a unique partition µ i such that J ci − /J ci ∼ = C ( T P i u ; End V i ) W i ∼ = M n ! /µ i ! C (cid:0) T σ ( µ i ) u (cid:14) Z S n ( σ ( µ i )) (cid:1) The induced homomorphism f i : φ − ( J ci − ) /φ − ( J ci ) → J i − /J i is injective. If we would know that K ∗ ( f i ) is an isomorphism for every i , then thetheorem would follow with Lemma 2.3.We know already that the number of components of Prim( C ∗ R ( R , q )) equalsthe number of components of Prim( C ∗ r ( W )) ∼ = (cid:102) T u /W . Therefore it suffices toconstruct, for every i , a projection p ∈ φ − ( J ci − ) /φ − ( J ci ) such that f i ( p ) ∈ J ci − /J ci has rank one.We can do this because we know precisely what all discrete series look like. Wemay assume that the central character of ( δ i , V i ) lies in T rs , so that the centralcharacter of π ( P i , δ i , t ) ◦ φ is W t . As a W -representation π ( P i , δ i , t ) ◦ φ isequivalent with Ind W W Pi ( (cid:15) W Pi ), where (cid:15) W P denotes the sign representation of W P ..6. A n − φ − ( J ci − ) does not annihilate this representation, but it is contained inthe kernel of the representation π ( P, δ, t (cid:48) ) for P ⊃ P i = P µ i and t (cid:48) ∈ T Pu ⊂ T P i = T σ ( µ i ) . As W -representation we have π ( P, δ, t (cid:48) ) ◦ φ ∼ = Ind W W P ( (cid:15) W P ). Hence the ”stalk” of φ − ( J ci − ) /φ − ( J ci ) at W t contains (cid:92) P (cid:41) P i ,t ∈ T Pu ker Ind W W P ( (cid:15) W P ) (cid:14) ker Ind W W Pi ( (cid:15) W Pi )which is a subquotient algebra of C [ W ]. Therefore we can find a suitable p alreadyin C [ W ]: pick a projection of minimal rank in C [ W ] which is in (cid:92) P (cid:41) P i ker Ind W W P ( (cid:15) W P ) , but not in ker Ind W W Pi ( (cid:15) W Pi ) . Then φ ( p ) will act as a rank one projector on π ( P i , δ i , t ) . (cid:50) A n − A affineHecke algebras does not depend on q , but it will still be insightful to determine thespectra of these algebras. The title of this section is A n − instead of A n becausewe want to consider everything as a quotient or a subset of Z n . If we requirethat our root datum is semisimple, then the easiest case is when X is the weightlattice. This is completely analogous to the GL n -case, we can even show in thesame way that Conjecture 5.28 holds. The calculations are also manageable when X is the root lattice. Intermediate lattices however would require quite some extrabookkeeping, so we do not study those. Throughout this section we assume that n >
2, because the root system A has slightly different properties.10 Chapter 6. Examples and calculationsThe root datum R ( A n − ) ∨ is defined as X = Z n / Z ( e + · · · e n ) ∼ = Q + (( e + · · · + e n ) /n − e n ) Q = { x ∈ Z n : x + · · · + x n = 0 } X + = { x ∈ X : x ≥ x ≥ · · · ≥ x n } Y = Q ∨ = { y ∈ Z n : y + · · · + y n = 0 } T = { t ∈ ( C × ) n : t · · · t n = 1 } t = ( t ( e ) , . . . , t ( e n )) = ( t , . . . , t n ) R = R = { e i − e j ∈ X : i (cid:54) = j } R ∨ = R ∨ = { e i − e j ∈ Y : i (cid:54) = j } F = { α i = e i − e i +1 } α = e − e n s i = s α i s = t α s α = t − α s α t α : x → x + α − (cid:104) α ∨ , x (cid:105) α W = (cid:104) s , · · · , s n − | s i = ( s i s i +1 ) = ( s i s j ) = e if | i − j | > (cid:105) ∼ = S n S aff = { s , s , . . . , s n − } W aff = (cid:104) s , W | s = ( s s i ) = ( s s ) = ( s s n − ) = e if 2 ≤ i ≤ n − (cid:105) W = W aff (cid:111) Ω Ω = (cid:104) t e − ( e + ··· e n ) /n (12 · · · n ) (cid:105) ∼ = Z /n Z Because all roots are conjugate, s is conjugate to any s i ∈ S aff , and for any labelfunction q ( s ) = q ( s i ) = q α ∨ i = q The W -stabilizer of (cid:0) ( t ) µ ( t µ +1 ) µ · · · ( t n ) µ d (cid:1) is isomorphic to S µ × · · · × S µ d .Generically there are n ! n residual points, and they all satisfy t ( α i ) = q or t ( α i ) = q − for 1 ≤ i < n . There residual points form n conjugacy classes unless q = 1, inwhich case T itself is the only residual coset. • group case q = According to (6.2) we have K ∗ (cid:0) C ∗ r ( W ) (cid:1) ⊗ C ∼ = ˇ H ∗ (cid:0)(cid:102) T u (cid:14) S n ; C (cid:1) ∼ = (cid:77) µ (cid:96) n ˇ H ∗ (cid:0) T σ ( µ ) u (cid:14) Z S n ( σ ( µ )); C (cid:1) Pick a partition µ of n and write it as in (6.65). T σ ( µ ) = { ( t ) µ ( t µ +1 ) µ · · · ( t n ) µ d ∈ T }∼ = { ( t ) µ ( t µ +1 ) µ · · · ( t n ) µ d ∈ ( C × ) n } / C × × { ( e πik/n ) n : 0 ≤ k < g } T σ ( µ ) (cid:14) Z S n ( σ ( µ )) ∼ = (( C × ) m n /S m n × · · · × ( C × ) m /S m ) (cid:14) C × ×{ ( e πik/n ) n : 0 ≤ k < g } where C × acts diagonally. By Lemma 6.1 each factor ( C × ) m i /S m i is homotopyequivalent to a circle. The induced action of S ⊂ C × on this direct productof circles identifies with a direct product of rotations. Hence T σ ( µ ) /Z S n ( σ ( µ ))is homotopy equivalent with T b ( µ ) − × { gcd( µ ) points } . The cohomology of thisspace has no torsion, so by (6.3) K ∗ (cid:0) C ∗ r ( W ) (cid:1) ∼ = ˇ H ∗ (cid:0)(cid:102) T u (cid:14) S n ; Z (cid:1) ∼ = Z d ( n ) d ( n ) = (cid:88) µ (cid:96) n gcd( µ )2 b ( µ ) − (6.73).6. A n − • generic, equal label case q (cid:54) = 1 • P µ = F \ { α µ , α µ + µ , . . . , α n − µ d } Inequivalent subsets of F are parametrized by partitions µ of n . For the typicalpartition (6.65) we put R P µ ∼ = ( A n − ) m n × · · · × ( A ) m ∼ = R ∨ P µ X P µ ∼ = (cid:0) Z ( e + · · · + e µ ) /µ + · · · + Z ( e n +1 − µ d + · · · + e n ) /µ d (cid:1)(cid:14) Z ( e + · · · + e n ) /gX P µ ∼ = (cid:0) Z n / Z ( e + · · · + e n ) (cid:1) m n × · · · × (cid:0) Z / Z ( e + e ) (cid:1) m Y P µ = { y ∈ Z ( e + · · · + e µ ) + · · · + Z ( e n +1 − µ d + · · · + e n ) : y + · · · + y n = 0 } Y P µ = { y ∈ Y : y + · · · + y µ = · · · = y n +1 − µ d + · · · + y n = 0 } T P µ = { ( t ) µ · · · ( t n ) µ d ∈ T : t µ /g · · · t µ d /gn = 1 } T P µ = { t ∈ T : t t · · · t µ = · · · = t n +1 − µ d · · · t n = 1 } K P µ = { t ∈ T P µ : t µ = · · · = t µ d n = 1 } W P µ ∼ = ( S n ) m n × · · · × ( S ) m W ( P µ , P µ ) ∼ = S m n × · · · × S m × S m W P µ P µ = K P µ (cid:111) W ( P µ , P µ ) Z S n ( σ ( µ )) = W ( P µ , P µ ) (cid:110) n (cid:89) l =1 ( Z /l Z ) m l The W P µ -orbits of residual points for H P µ are represented by the points (cid:0) ( q ( µ − / , q ( µ − / , . . . , q (1 − µ ) / ) · · · ( q ( µ d − / , q ( µ d − / , . . . , q (1 − µ d ) / ) (cid:1) · (cid:0) ( e πik /µ ) µ · · · ( e πik d /µ d ) µ d (cid:1) , ≤ k i < µ i (6.74)These points are in bijection with K P µ × Z / gcd( µ ) Z . Also T σ ( µ ) consists of exactlygcg( µ ) components, one of which is T P µ . Together with Proposition 3.20.2 thisleads to (cid:91) ∆ Pµ (cid:0) P µ , δ, T P µ (cid:1)(cid:14) K P µ ∼ = T P µ × Z / gcd( µ ) Z ∼ = T σ ( µ ) (cid:91) ∆ Pµ (cid:0) P µ , δ, T P µ (cid:1)(cid:14) W P µ P µ ∼ = T σ ( µ ) (cid:14) Z S n ( σ ( µ ))If a point t ∈ T P µ has a nontrivial stabilizer in W ( P µ , P µ ) ∼ = (cid:81) nl =1 S m l thenthis stabilizer is generated by transpositions. From (6.67) we see that every suchtransposition w ∈ W ( P µ , P µ ) can be written as a product of mutually commutingreflections s α with α ∈ R and c − α ( t ) = 0. So by (3.122) ı ow ( t ) = 1 and since thediscrete series representation δ is onedimensional, also π ( w, P µ , δ, t ) = 1. On theother hand, K P µ permutes the components of (cid:83) ∆ Pµ (cid:0) P µ , δ, T P µ (cid:1) faithfully, so theaction of W P µ P µ on12 Chapter 6. Examples and calculations C (cid:18) (cid:91) ∆ Pµ T P µ ; M n ! /µ ! ( C ) (cid:19) is essentially only on the underlying space. Therefore C ∗ r ( R , q ) ∼ = (cid:77) µ (cid:96) n M n ! /µ ! (cid:16) C (cid:16) (cid:71) ∆ Pµ T P µ u (cid:17)(cid:17) ∼ = (cid:77) µ (cid:96) n M n ! /µ ! (cid:0) T σ ( µ ) u (cid:14) Z S n ( σ ( µ )) (cid:1) K ∗ (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = (cid:77) µ (cid:96) n K ∗ (cid:0) T σ ( µ ) u (cid:14) Z S n ( σ ( µ )) (cid:1) ∼ = (cid:77) µ (cid:96) n K ∗ (cid:0) T b ( µ ) − (cid:1) ∼ = Z d (6.75)where d = (cid:80) µ (cid:96) n gcd( µ )2 b ( µ ) − .We conclude that for R ( A n − ) ∨ the K -theory of C ∗ r ( R , q ) does not depend on q , and is a free abelian group. Theorem 6.3 K ∗ ( φ ) : K ∗ (cid:0) C ∗ r (cid:0) R ( A n − ) ∨ , q (cid:1)(cid:1) → K ∗ (cid:0) C ∗ r (cid:0) R ( A n − ) ∨ , q (cid:1)(cid:1) is an isomorphism.Proof. This is completely analogous to Theorem 6.2. The essential common prop-erties of these two root data are that the reduced C ∗ -algebras are Morita equivalentto commutative C ∗ -algebras, and that we have explicit descriptions of the discreteseries representations of the algebras H P . (cid:50) The analogy with R ( GL n ) is significantly weaker for the root datum R ( A n − ): X = Q = { x ∈ Z n : x + · · · + x n = 0 } X + = { x ∈ X : x ≥ x ≥ · · · ≥ x n } Q ∨ = { y ∈ Z n : y + · · · + y n = 0 } Y = Z n / Z ( e + · · · + e n ) ∼ = Q ∨ + (( e + · · · + e n ) /n − e ) T = ( C × ) n / C × t = ( t , . . . , t n ) = ( t ( e ) , . . . , t ( e n )) R = R = { e i − e j ∈ X : i (cid:54) = j } R ∨ = R ∨ = { e i − e j ∈ Y : i (cid:54) = j } F = { α i = e i − e i +1 : 1 ≤ i < n } α = e − e n s i = s α i s = t α s α = t − α s α t α : x → x + α − (cid:104) α ∨ , x (cid:105) α W = (cid:104) s , · · · , s n − | s i = ( s i s i +1 ) = ( s i s j ) = e if | i − j | > (cid:105) ∼ = S n S aff = { s , s , . . . , s n − } Ω = { e } W = W aff = (cid:104) s , W | s = ( s s i ) = ( s s ) = ( s s n − ) = e if 2 ≤ i ≤ n − (cid:105) q ( s ) = q ( s i ) = q α ∨ i := q For q (cid:54) = 1 there are n ! residual points. They form one W -orbit, and a typicalresidual point is.6. A n − (cid:0) q (1 − n ) / , q (3 − n ) / , . . . , q ( n − / (cid:1) To determine the isotropy group of points of T we have to be careful. In generalthe W -stabilizer of (cid:0) ( t ) µ ( t µ +1 ) µ · · · ( t n ) µ d (cid:1) ∈ T is isomorphic to S µ × S µ × · · · × S µ d ⊂ W However, in some special cases the diagonal action of C × on ( C × ) n gives rise toextra stabilizers. Let r be a divisor of n, k ∈ ( Z /r Z ) × and λ = ( λ , . . . , λ l ) apartition of n/r . The isotropy group of (cid:0) ( t ) λ ( e πik/r t ) λ · · · ( e − πik/r t ) λ ( t rλ +1 ) λ · · · ( e − πik/r t rλ +1 ) λ · · · ( e − πik/r t n ) λ l (cid:1) (6.76)is isomorphic to S rλ × S rλ × · · · × S rλ l (cid:111) Z /r Z (6.77)Explicitly the subgroup Z /r Z is generated by(1 λ +1 2 λ +1 · · · ( r − λ +1)(2 λ +2 2 λ +2 · · · ( r − λ +2) · · · ( λ λ · · · rλ ) · · · ( n +1 − rλ d n +1+(1 − r ) λ d · · · n +1+( r − λ d )( n +(1 − r ) λ d n +(2 − r ) λ d · · · n )(6.78)and it acts on every factor S rλ j in (6.77) by cyclic permutations. • group case q = As we saw before K ∗ (cid:0) C ∗ r ( W ) (cid:1) ⊗ C ∼ = ˇ H ∗ (cid:0)(cid:102) T u (cid:14) S n ; C (cid:1) ∼ = (cid:77) µ (cid:96) n ˇ H ∗ (cid:0) T σ ( µ ) u (cid:14) Z S n ( σ ( µ )); C (cid:1) For the typical partition µ we have T σ ( µ ) = { ( t ) µ ( t µ +1 ) µ · · · ( t n ) µ d } (cid:14) C × ×{ t : t ( e j ) = e πijk/g , ≤ k < g } (6.79)which is the disjoint union of g = gcd( µ ) complex tori of dimension m n + m n − + · · · + m − T σ ( µ ) (cid:14) Z S n ( σ ( µ )) ∼ = (cid:0) ( C × ) m n /S m n × · · · × ( C × ) m /S m (cid:1) (cid:14) C × ×{ t : t ( e j ) = e πijk/g , ≤ k < g } (6.80)Curiously enough these sets are diffeomorphic to the corresponding sets for R ( A n − ) ∨ ,a phenomenon for which the author does not have a good explanation. Anyway,14 Chapter 6. Examples and calculationswe do take advantage of this by reusing our deduction that (6.80) is homotopyequivalent with T b ( µ ) − × { gcd( µ ) points } . Just as in (6.73) we conclude that K ∗ (cid:0) C ∗ r ( W ) (cid:1) ∼ = ˇ H ∗ (cid:0)(cid:102) T u (cid:14) S n ; Z (cid:1) ∼ = Z d ( n ) d ( n ) = (cid:88) µ (cid:96) n gcd( µ )2 b ( µ ) − (6.81) • generic case q (cid:54) = This is noticeably different from the generic cases for R ( GL n ) and R ( A ∨ n − ) be-cause C ∗ r ( R ( A n − , q )) is not Morita equivalent to a commutative C ∗ -algebra. Ofcourse the inequivalent subsets of F are still parametrized by partitions µ of n . • P µ = F \ { α µ , α µ + µ , . . . , α n − µ d } R P µ ∼ = ( A n − ) m n × · · · × ( A ) m ∼ = R ∨ P µ X P µ ∼ = { x ∈ Z ( e + · · · + e µ ) /µ + · · · + Z ( e n +1 − µ d + · · · + e n ) /µ d : x + · · · + x n = 0 } X P µ ∼ = { x ∈ Z µ / Z ( e + · · · + e µ ) + · · · + Z µ d / Z ( e n +1 − µ d + · · · + e n ) : x + · · · + x n ∈ g Z /g Z } Y P µ ∼ = Z ( e + · · · + e µ ) + · · · + Z ( e n +1 − µ d + · · · + e n ) / Z ( e + · · · + e n ) Y P µ ∼ = { y : y + · · · + y µ = · · · = y n +1 − µ d + · · · + y n = 0 } / Z ( e + · · · e n ) T P µ = { ( t ) µ · · · ( t n ) µ d } / C × T P µ = { t : t t · · · t µ = · · · = t n +1 − µ d · · · t n = 1 } / { z ∈ C : z g = 1 } K P µ = { ( t ) µ · · · ( t n ) µ d : t µ = · · · = t µ d n = 1 } / { z ∈ C : z g = 1 } W P µ ∼ = S m n n × S m n − n − × · · · × S m W ( P µ , P µ ) ∼ = S m n × · · · × S m × S m Note that T σ ( µ ) = T P µ × { t : t ( e j ) = e πijk/g , ≤ k < g } The W P µ -orbits of residual points for H P µ are represented by the points of K P µ (cid:0) q ( µ − / , q ( µ − / , . . . q (1 − µ ) / , q ( µ − / , . . . , q ( µ d − / , . . . , q (1 − µ d ) / (cid:1) Hence the intertwiners π ( k ) with k ∈ K P µ permute the elements of ∆ P µ faithfully,and (cid:71) ∆ Pµ (cid:0) P µ , δ, T P µ (cid:1) /K P µ ∼ = T P µ = (cid:0) T σ ( µ ) (cid:1) where ( · ) means the connected component containing (1 , , . . . ,
1) = 1 ∈ T . In(6.70) we saw that the intertwiners for R ( GL n ) , q (cid:54) = 1 have the property w ( t ) = t ⇒ π ( w, P µ , δ, t ) = 1This implies that in our present setting we can have w ( t ) = t and π ( w, P µ , δ, t ) (cid:54) = 1only if w ( t ) = t does not hold without taking the action of C × into account. Let.6. A n − w ∈ W ( P µ , P µ ) and t ∈ T P µ up to conjugacy. For a divisor r of g ∨ := gcd( µ ∨ ) we have the partition µ /r := ( nr ) m n /r · · · (2 r ) m /r ( r ) m /r Notice that b ( µ /r ) = b ( µ ) = b ( µ ∨ )There exists a σ ∈ S n which is conjugate to σ ( µ /r ) and satisfies σ r = σ ( µ ). Weconstruct a particular such σ as follows. If r = g ∨ then (starting from the left)replace every block( d + 1 d + 2 · · · d + m )( d + 1 + m · · · d + 2 m ) · · · ( d + ( g ∨ − m · · · d + g ∨ m )of σ ( µ ) by( d +1 d +1+ m · · · d +1+( g ∨ − m d +2+ m · · · d +2+( g ∨ − m d +3 · · · d + g ∨ m )We denote the resulting element by σ ( µ ) /g ∨ and for general r | g ∨ we define σ ( µ ) /r := (cid:0) σ ( µ ) /g ∨ (cid:1) g ∨ /r Consider the cosets of subtori T P µ r,k := (cid:0) T σ ( µ ) /r (cid:1) (cid:0) (1) g ∨ µ /r ( e πik/r ) g ∨ µ g ∨ /r /r · · · ( e − πik/r ) g ∨ µ d /r (cid:1) k ∈ Z If gcd( k, r ) = 1 then the generic points of T P µ r,k have W ( P µ , P µ )-stabilizer (cid:104) W P µ , σ ( µ ) /r (cid:105) ∩ W ( P µ , P µ ) ∼ = Z /r Z Note that for r (cid:48) | g ∨ T P µ r (cid:48) ,k ⊂ T P µ r,k if r | r (cid:48) (6.82)If a point t ∈ T P µ r,k does not lie on any T P µ r (cid:48) ,k (cid:48) with r (cid:48) > r , then its W ( P µ , P µ )-stabilizer may still be larger than Z /r Z . However, it is always of the form S rλ × · · · × S rλ l (cid:111) Z /r Z By an argument like on pages 207 and 211 one can show that the intertwiners π ( w, P µ , δ, t ) are scalar for w ∈ S rλ × · · · × S rλ l and nonscalar for w ∈ ( Z /r Z ) \ { e } .Because Z /r Z is cyclic this implies that π ( P µ , δ, t ) is the direct sum of exactly r inequivalent irreducible representations. For a more systematic discussion of suchmatters we refer to [40].Different choices of σ ( µ ) /r or of k ∈ ( Z /r Z ) × lead to conjugate subvarietiesof T P µ , so we have a complete discription of Prim (cid:0) C ∗ r ( R , q ) P µ (cid:1) . To calculate the K -theory of this algebra we use Theorem 2.24, which says that (modulo torsion)it is isomorphic to H ∗ W ( P µ ,P µ ) (cid:0) T P µ u ; L u (cid:1) ∼ = ˇ H ∗ (cid:0) T P µ (cid:14) W ( P µ , P µ ); L W ( P µ ,P µ ) u (cid:1)
16 Chapter 6. Examples and calculationsWe know from [62] that we can endow T P µ u with the structure of a finite W ( P µ , P µ )-CW-complex, such that every T P µ u,r,k is a subcomplex. The local coefficient system L u is not very complicated: L u ( B ) ∼ = Z r if and only if B \ ∂B consists of genericpoints in a conjugate of T P µ u,r,k . In suitable coordinates the maps L u ( B → B (cid:48) ) areall of the form Z r → Z r/d : ( x , . . . , x r ) → ( x + x + · · · + x d , . . . , x r − d + · · · + x r )Hence the associated sheaf is the direct sum of several subsheaves F µr , one for eachdivisor r of gcd( µ ∨ ). The support of F µr is W ( P µ , P µ ) T P µ u,r, (cid:14) W ( P µ , P µ ) ∼ = T P µ /r u (cid:14) Z S n ( σ ( µ /r ))and on that space it has constant stalk Z φ ( r ) . Here φ is the Euler φ -function, i.e. φ ( r ) = { m ∈ Z : 0 ≤ m < r : gcd( m, r ) = 1 } = Z /r Z ) × This the rank of F µr because in every point of T u,r, we have r irreducible repre-sentations, but the ones corresponding to numbers that are not coprime with r are already accounted for by the sheaves F µr (cid:48) with r (cid:48) | r . Now we can calculateˇ H ∗ (cid:0) T P µ /W ( P µ , P µ ); L W ( P µ ,P µ ) u (cid:1) ∼ = (cid:77) r | gcd( µ ∨ ) ˇ H ∗ (cid:0) T P µ /W ( P µ , P µ ); F µr (cid:1) ∼ = (cid:77) r | gcd( µ ∨ ) ˇ H ∗ (cid:0) T P µ /r u (cid:14) Z S n ( σ ( µ /r )); Z φ ( r ) (cid:1) ∼ = (cid:77) r | gcd( µ ∨ ) ˇ H ∗ (cid:0) T b ( µ /r ) − ; Z φ ( r ) (cid:1) ∼ = (cid:77) r | gcd( µ ∨ ) Z φ ( r )2 b ( µ /r ) − = (cid:77) r | gcd( µ ∨ ) Z φ ( r )2 b ( µ ∨ ) − = Z gcd( µ ∨ )2 b ( µ ∨ ) − (6.83)By Theorem 2.24 K ∗ (cid:0) C ∗ r ( R , q ) P µ (cid:1) must also be a free abelian group of rankgcd( µ ∨ )2 b ( µ ∨ ) − .Summing over partitions µ of n we find that K ∗ (cid:0) C ∗ r ( R , q ) (cid:1) is a free abeliangroup of rank (cid:88) µ (cid:96) n gcd( µ ∨ )2 b ( µ ∨ ) − = (cid:88) µ (cid:96) n gcd( µ )2 b ( µ ) − We conclude that for the root datum R ( A n − ) K ∗ (cid:0) C ∗ r ( R , q ) (cid:1) ∼ = K ∗ (cid:0) C ∗ r ( W ) (cid:1) (6.84).7. B n B n The root systems of type B n are more complicated than those of type A n becausethere are roots of different lengths. This implies that the associated root data allowlabel functions which have three independent parameters. Detailed informationabout the representations of type B n affine Hecke algebras is available from [121].We will compare the C ∗ -algebras for generic labelled root data with the reduced C ∗ -algebra of the affine Weyl group of type B n . Consider the root datum R ( B n )where X is the root lattice: X = Q = Z n X + = { x ∈ X : x ≥ x ≥ · · · ≥ x n ≥ } Y = Z n Q ∨ = { y ∈ Y : y + · · · + y n even } T = ( C × ) n t = ( t , . . . , t n ) = ( t ( e ) , . . . , t ( e n )) R = { x ∈ X : (cid:107) x (cid:107) = 1 or (cid:107) x (cid:107) = √ } R = { x ∈ X : (cid:107) x (cid:107) = 2 or (cid:107) x (cid:107) = √ } R ∨ = { x ∈ X : (cid:107) x (cid:107) = 2 or (cid:107) x (cid:107) = √ } R ∨ = { x ∈ X : (cid:107) x (cid:107) = 1 or (cid:107) x (cid:107) = √ } F = { α i = e i − e i +1 : i = 1 , . . . , n − } ∪ { α n = e n } α = e s i = s α i s = t α s α : x → x + (cid:104) α ∨ , x (cid:105) α W = (cid:104) s , . . . , s n | s j = ( s i s j ) = ( s i s i +1 ) = ( s n − s n ) = e : i ≤ n − , | i − j | > (cid:105) S aff = { s , s , . . . , s n − , s n } Ω = { e } W = W aff = (cid:104) W , s | s = ( s s i ) = ( s s ) = e : i ≥ (cid:105) For a generic label function we have different labels q = q ( s ) ,q = q ( s i ) , ≤ i < n and q = q ( s n ). For completeness we mention that q α ∨ i = q q α ∨ n = q q α ∨ n / = q q − The finite reflection group W = W ( B n ) is naturally isomorphic to ( Z / Z ) n (cid:111) S n .If µ (cid:96) n then the W -stabilizer of (cid:0) (1) µ ( − µ ( t µ + µ +1 ) µ · · · ( t n ) µ d (cid:1) ∈ T is isomorphic to W ( B µ ) × W ( B µ ) × S µ × S µ d • group case q = q = q = In view of (6.2) we want to determine the extended quotient (cid:101)
T /W . Therefore westart with the classification of conjugacy classes in W . We already know that thequotient of W by the normal subgroup ( Z / Z ) n of sign changes is isomorphic to S n , and that conjugacy classes in S n are parametrized by partitions of n . So wewonder what the different conjugacy classes in ( Z / Z ) n σ ( µ ) are for µ (cid:96) n .18 Chapter 6. Examples and calculationsTo handle this we introduce the some notations. Assume that | µ | + | λ | = n and | µ | + | λ | + | ρ | = n (cid:48) . ν I = (cid:81) i ∈ I s e i I ⊂ { , . . . , n } I λ = { , λ , λ + λ , · · · } λ = ( λ , λ , λ , . . . ) σ (cid:48) ( λ ) = ν I λ σ ( λ ) ∈ W ( B | λ | ) σ ( µ, λ ) = σ ( µ ) ( m → m − | λ | mod n ) σ (cid:48) ( λ ) ( m → m + | λ | mod n ) σ ( µ, λ, ρ ) = σ ( µ, λ ) ( m → m − | ρ | mod n (cid:48) ) σ (cid:48) ( ρ ) ( m → m + | ρ | mod n (cid:48) )(6.85)Let I ⊂ { , . . . , m } and J ⊂ { m + 1 , . . . , m } . It is easily verified that ν I (1 2 · · · m )is conjugate to µ J ( m + 1 m + 2 · · · m ) if and only if | I | + | J | is even. Therefore theconjugacy classes in W are parametrized by ordered pairs of partitions of totalweight n . Explicitly ( µ, λ ) corresponds to σ ( µ, λ ) as in (6.85). The set T σ ( µ,λ ) and the group Z W ( B n ) ( σ ( µ, λ )) are both the direct product of the correspondingobjects for the blocks of µ and λ , i.e. for the parts ( m, m, . . . , m ). The centralizer of σ (( m ) k ) in W ( B km ) is generated by (1 2 · · · m ) , ν { , ,...,m } and the transpositionsof cycles.( am + 1 am + m + 1)( am + 2 am + m + 2) · · · ( am + m am + 2 m ) 0 ≤ a ≤ k − Z W ( B km ) ( σ (( m ) k )) ∼ = W ( B k ) (cid:0) ( C × ) km (cid:1) σ (( m ) k ) = (cid:8)(cid:0) ( t ) m ( t m +1 ) m · · · ( t km +1 − m ) m (cid:1) : t i ∈ C × (cid:9)(cid:0) T km (cid:1) σ (( m ) k ) (cid:14) Z W ( B km ) ( σ (( m ) k )) ∼ = [ − , k (cid:14) S k (6.87)Now consider the following element of W ( B km ): σ (cid:48) (( m ) k ) = ν { ,m +1 ,...,km +1 − m } (1 2 · · · m )( m + 1 · · · m ) · · · ( km + 1 − m · · · km )It has only 2 k fixpoints, namely (cid:0) ( ± m ( ± m · · · ( ± m (cid:1) The centralizer of σ (cid:48) (( m ) k ) is generated by ν { } (1 2 · · · m ) , ν { , ,...,m } and theelements (6.86). Hence Z W ( B mk ) ( σ (cid:48) (( m ) k )) ∼ = W ( B k ) (cid:0) T km (cid:1) σ (cid:48) (( m ) k ) (cid:14) Z W ( B mk ) ( σ (cid:48) (( m ) k )) ∼ = { (1) am ( − ( k − a ) m : 0 ≤ a ≤ k } (6.88)Now we can see what T σ ( µ,λ ) u (cid:14) Z W ( σ ( µ, λ )) looks like. Its number of components N ( λ ) depends only on λ , and all these components are mutually homeomorphiccontractible orbifolds, the shape and dimension being determined by µ . Moreprecisely, for every block of µ of width k we get a factor [ − , k /S k , and forevery block of λ of width l we must multiply the number components by l + 1..7. B n W ) as T σ ( µ,λ ) u (cid:14) Z W ( σ ( µ, λ )) = (cid:71) λ ∪ λ = λ (cid:0) T σ ( µ,λ ,λ ) u (cid:14) Z W ( σ ( µ, λ , λ )) (cid:1) c = (cid:71) λ ∪ λ = λ (cid:0) T | µ | (cid:1) σ ( µ ) (cid:14) Z W ( B | µ | ) ( σ ( µ )) ( − | λ | (1) | λ | (6.89)where the subscript c is supposed to indicate that we take only the connectedcomponent containing the point (cid:0) (1) | µ | ( − | λ | (1) | λ | (cid:1) .In effect we parametrized the components of the extended quotient (cid:102) T u /W byordered triples of partitions ( µ, λ , λ ) of total weight n , and every such compo-nents is contractible. Denote the number of ordered k -tuples of partitions of totalweight n by P ( k, n ).ˇ H ∗ (cid:0)(cid:102) T u /W ; Z (cid:1) = ˇ H (cid:0)(cid:102) T u /W ; Z (cid:1) ∼ = Z P (3 ,n ) By (6.3) also K ∗ (cid:0) C ∗ r ( W ) (cid:1) = K (cid:0) C ∗ r ( W ) (cid:1) ∼ = Z P (3 ,n ) (6.90) • generic case • P µ = F \ { α µ , α µ + µ , . . . , α | µ | } The inequivalent subsets of F are parametrized by partitions µ of weight at most n . R P µ ∼ = ( A n − ) m n × · · · × ( A ) m × B n −| µ | R ∨ P µ ∼ = ( A n − ) m n × · · · × ( A ) m × C n −| µ | X P µ ∼ = Z ( e + · · · + e µ ) /µ + · · · + Z ( e | µ | +1 − µ d + · · · + e | µ | ) /µ d X P µ ∼ = ( Z n / Z ( e + · · · + e n )) m n × · · · × ( Z / ( Z ( e + e )) m × Z n −| µ | Y P µ = Z ( e + · · · + e µ ) + · · · + Z ( e | µ | +1 − µ d + · · · + e | µ | ) Y P µ = { y ∈ Z n : y + · · · + y µ = · · · = y | µ | +1 − µ d + · · · + y | µ | = 0 } T P µ = { ( t ) µ ( t µ +1 ) µ · · · ( t | µ | ) µ d (1) n −| µ | : t i ∈ C × } T P µ = { t ∈ ( C × ) n : t · · · t µ = t µ · · · t µ + µ = · · · = t | µ | +1 − µ d · · · t | µ | = 1 } K P µ = { t ∈ T P µ : t µ = · · · = t µ d | µ | = 1 } W P µ ∼ = S m n n × · · · × S m × W ( B n −| µ | ) W ( P µ , P µ ) ∼ = W ( B m n ) × · · · × W ( B m )We see that R P µ is the product of various root data of type A m and one factor R ( B n −| µ | ). Hence by (3.42) H P µ is the tensor product of a type A part anda type B part. From our study of R ( A n − ) we recall that the discrete seriesrepresentations of the type A part of H P µ are in bijection with K P µ . From [58,20 Chapter 6. Examples and calculationsProposition 4.3] and [98, Appendix A.2] we know that the residual points for B n −| µ | are parametrized by ordered pairs ( λ , λ ) of total weight n − | µ | . Theunitary part of such a residual point is in the component we indicated in (6.89).Let RP ( R , q ) denote the collection of residual points for the pair ( R , q ). (cid:71) t ∈ RP ( R Pµ ,q Pµ ) tT P µ u (cid:14) W P µ P µ ∼ = (cid:71) t ∈ RP ( R ( B n −| µ | ,q )) tT P µ u (cid:14) W ( P µ , P µ ) ∼ = T P µ u (cid:14) Z W ( B | µ | ) ( σ ( µ )) × (cid:71) ( λ ,λ ): | λ | + | λ | = n ( − | λ | (1) | λ | (6.91)This space is diffeomorphic to the extended quotient described on page 219. ByTheorem 3.19 every point is the central character of at least one irreducible C ∗ r ( R , q )-representation. Therefore it is natural to compare Conjecture 5.24 withthe following statements. Every parabolically induced C ∗ r ( R ( B n ) ∨ , q )-representation π ( P, δ, t ) is irre-ducible. Prim (cid:0) C ∗ r ( R ( B n ) ∨ , q ) (cid:1) is naturally in bijection with (6.91).Opdam and Slooten [99] have announced a proof of 1). Extending the resultsfrom [122] they can show that all R-groups are trivial in this situation. In viewof the above calculations and (5.82), 1) and Theorem 5.25 together imply 2).Probably 2) can also be derived from the recent work of Kato [74, 75]. Theorem 6.4
Let q be a generic positive label function on the root datum R ( B n ) .Then Conjecture 5.24 holds. In particular there are precisely P (2 , n ) inequivalentdiscrete series representations, one for each W -orbit of residual points.Proof. This theorem was predicted in [99, § W t ∈ T u /W the image of G ( φ W t ) is spanned by the modules G ( φ ) π ( P, W P r, δ, t ) = π ( P, W P r u , ˜ σ ( δ ) , t ) with ( P, δ ) ∈ ∆ , W r u t = W t (6.92)By 2) the number of such modules equals the number of irreducible W -representations with central character W t . So we only have to show that theelements (6.92) are linearly independent in G ( C ∗ r ( W )). Abbreviate E = Rep W t (cid:0) C ∗ r ( W ) (cid:1) ⊗ Z C Fix a set P (cid:48) of representatives for the action of W on the power set of F . For P, Q ∈ P (cid:48) we write P ≤ Q if ∃ w ∈ W : wP ⊂ Q .7. B n t ∈ W t such that P := { α ∈ F : θ α ( t ) = 1 } is maximal. We may assume that the set P of short roots corresponding to P is an element of P (cid:48) . From the proof of Theorem 3.15 we see that the standardpairing between representations of the isotropy group W ,t gives an inner producton E .Now we can decompose E into subspaces corresponding to the P ∈ P (cid:48) . Let E P be the span of the virtual representations in E whose character only allowscontinuous deformations along T Pu /W , not along other directions on T u /W . Weput E P = E P ∩ (cid:16) (cid:88) Q>P E Q (cid:17) ⊥ From Theorem 3.15, (6.89), (6.91) and (2.106) (for HH ) we deducedim E P = { ( P, W P r, δ, t ) ∈ Ξ u : W r u t = W t } (6.93)Notice that this is a simple version of [99, Proposition 6.6]. Upon applying (5.82)to the affine Hecke algebra H P (which has generic labels), we see that G ( φ ) ⊗ id C (cid:0) span { ( P, W P r, δ, t ) ∈ Ξ u : W r u t = W t } (cid:1) ⊂ E P (6.94)and that G ( φ ) ⊗ id C is injective on this domain. Moreover by (6.93) the image in(6.94) intersects (cid:80) Q>P E Q only in 0. Therefore G ( φ ) is injective. (cid:50)
22 Chapter 6. Examples and calculations ppendix A
Crossed products
We collect some well-known results on crossed products of algebras by compactgroups. We do this in the large category of m -algebras, but it is straightforward tosee that they also hold for C ∗ -algebras, if we assume that the action is *-preserving.So let A be an m -algebra, G a compact group (with its normalized Haar measure)and α : G × A → Aα ( g, a ) = α g ( a ) (A.1)a continuous action of G on A by algebra homomorphisms. Recall that the crossedproduct A (cid:111) α G is the vector space C ( G ; A ) with multiplication defined by( f · f (cid:48) )( g (cid:48) ) = (cid:90) G f ( g ) α g ( f (cid:48) ( g − g (cid:48) )) dg (A.2)This is again an m -algebra. Notice that if a ∈ A and δ g is the δ -function con-centrated at g ∈ G , then aδ g is an element of the multiplier algebra M ( A (cid:111) α G ).Explicitly, multiplying by this element is defined as( f · aδ g )( g (cid:48) ) = f ( g (cid:48) g − ) α g (cid:48) g − ( a ) (A.3)( aδ g · f )( g (cid:48) ) = aα g ( f ( g − g (cid:48) )) (A.4)Similarly, if N is a closed subgroup of G then any ψ ∈ C ∗ ( N ) is also a multiplierof A (cid:111) α G , defined naturally by( f · ψ )( g (cid:48) ) = (cid:90) N f ( g (cid:48) n − ) ψ ( n ) dn (A.5)( ψ · f )( g (cid:48) ) = (cid:90) N ψ ( n ) α n ( f ( n − g (cid:48) )) dn (A.6)The next result is useful in connection with projective representations. Lemma A.1
Let { e } → N → G → H → { e } be a short exact sequence of compactgroups, all equipped with their normalized Haar measures. Assume that N ⊂ ker α , so that α descends to H . Let p N ∈ C ∗ ( N ) be the constant function with value 1,considered as an idempotent in M ( A (cid:111) α G ) .a) p N is central and p N ( A (cid:111) α G ) ∼ = ( A (cid:111) α H ) Assume now that moreover N is finite and the extension from H to G by N central.Then we let (cid:98) N be the dual of the abelian group N and we consider every (one-dimensional) character χ of N as an idempotent p χ ∈ M ( A (cid:111) α G ) . In particular p N = p χ for the trivial character χ ≡ .b) p χ is also central and A (cid:111) α G = (cid:76) χ ∈ b N p χ ( A (cid:111) α G ) Proof.
For any b ∈ A (cid:111) α G and g (cid:48) ∈ G we have( b · p χ )( g (cid:48) ) = (cid:90) G b ( g ) α g ( p χ ( g − g (cid:48) )) dg = (cid:90) N b ( g (cid:48) g − ) χ ( g − ) | N | − dg = (cid:90) N | N | − χ ( g − ) b ( g − g (cid:48) ) dg = (cid:90) G p χ ( g ) α g ( b ( g − g (cid:48) )) dg = ( p χ · b )( g (cid:48) ) (A.7)The third equality holds if N is central or if χ ≡ N only normal, that is,in all the cases we need. So p χ indeed commutes with A (cid:111) α G and p χ ( A (cid:111) α G ) isa subalgebra. Now the statement b) follows from two standard equalities in therepresentation theory of finite groups: (cid:88) χ ∈ ˆ N p χ = 1 and p χ · p χ (cid:48) = δ χχ (cid:48) p χ (A.8)Writing π : G → H , it is easily verified that π ∗ : A (cid:111) α H → A (cid:111) α G : f → f ◦ π (A.9)is a monomorphism of m -algebras, so let us determine its image. Since( p N · π ∗ f )( g (cid:48) ) = (cid:90) G p N ( g ) α g ( π ∗ f ( g − g (cid:48) )) dg = (cid:90) N p N ( g ) f ( π ( g − g (cid:48) )) dg = (cid:90) N | N | − f ( πg (cid:48) ) dg = π ∗ f ( g (cid:48) ) (A.10)the image is contained in p N ( A (cid:111) α G ). From the above expressions for bp N it fol-lows immediately that anything of this type is N -biinvariant, so that it descendsto an element of A (cid:111) α H . Hence the image of π ∗ is exactly p N ( A (cid:111) α G ) (cid:50) A is unital and we are given p ∈ C ( G ; A × ) with the properties p ( gg (cid:48) ) = p ( g ) α g ( p ( g (cid:48) )) (A.11) p ( e ) = 1 (A.12)Then p is an idempotent in A (cid:111) α G and we can define a new action β of G on A by β g ( a ) = p ( g ) α g ( a ) p ( g ) − (A.13)The following description of the invariant algebra A β ( G ) := { a ∈ A : β g ( a ) = a ∀ g ∈ G } (A.14)is essentially due to Rosenberg [109]. Lemma A.2 A β ( G ) ∼ = p ( A (cid:111) α G ) p Proof.
We will show that the obvious map φ : A β ( G ) → p ( A (cid:111) α G ) pφ ( a ) = p ( aδ e ) p (A.15)is an isomorphism. Clearly φ is linear and continuous. If a, a (cid:48) ∈ A β then φ ( a ) = p ( aδ e ) p = p ( aδ e ) = ( aδ e ) p (A.16) φ ( a ) φ ( a (cid:48) ) = p ( aδ e ) pp ( a (cid:48) δ e ) p = p ( aδ e )( a (cid:48) δ e ) p = p ( aa (cid:48) δ e ) p = φ ( aa (cid:48) ) (A.17)so φ turns out to be injective and multiplicative. Next we observe that for any g ∈ G p ( g ) δ g · p = p = p · p ( g ) δ g (A.18)This gives, for b ∈ p ( A (cid:111) α G ) p , b ( g ) = ( p ( g ) δ g · b )( g ) = p ( g ) α g ( b ( e )) (A.19) b ( g ) = ( b · p ( g ) δ g )( g ) = b ( e ) p ( g ) (A.20)Comparing these expressions we see that b ( e ) ∈ A β . Therefore φ is bijective, withcontinuous inverse b → b ( e ) . (cid:50) Assume now that G is finite. Then evaluating integrals of A -valued functionson G is easy, so we do not need any topology on A to define A (cid:111) α G . Moreover weagree to use the counting measure on G , even though it is not normalized. Thecrossed product thus obtained also appears in another way: Lemma A.3
Let ( C [ G ] , ρ ) be the right regular representation of G and A any com-plex algebra. Endow A ⊗ End( C [ G ] ⊗ C n ) with the G -action g ( a ) := ρ ( g ) α g ( a ) ρ ( g − ) ,where α stands also for the action of G on A tensored with the identity. Then (cid:0) A ⊗ End( C [ G ] ⊗ C n ) (cid:1) G ∼ = M n ( A (cid:111) α G )26 Appendix A. Crossed products Proof.
The left hand side is isomorphic to M n (cid:16)(cid:0) A ⊗ End( C [ G ]) (cid:1) G (cid:17) , so it sufficesto prove the case n = 1.For a ∈ A and g, h ∈ G define L ( a ⊗ g )( h ) = α h − g − ( a ) ⊗ gh . It is easily checkedthat this extends to an algebra homomorphism L : A (cid:111) α G → (cid:0) A ⊗ End( C [ G ]) (cid:1) G .We claim that L (cid:48) : b → (cid:80) g ∈ G b ( g − ) e ⊗ g is the inverse of L . It is clear that L (cid:48) L = Id, so we only have to show that L ( L (cid:48) b ) = b for any b ∈ (cid:0) B ⊗ End( C [ G ]) (cid:1) G . L ( L (cid:48) b )( h ) = (cid:88) g ∈ G α h − g − ( b ( g − ) e ) ⊗ gh = α h − α g − ( b )( g − ) e ⊗ gh = (cid:88) g ∈ G α h − (cid:0) ρ ( g ) bρ ( g − )( g − ) (cid:1) e ⊗ gh = (cid:88) g ∈ G α h − (cid:0) b ( e ) g − (cid:1) e ⊗ gh = α h − ( b )( e ) h = ρ ( h − ) α h − ( b ) ρ ( h )( h ) = b ( h ) (A.21)This holds for any h ∈ G , so indeed L (cid:48) = L − (cid:50) ibliography [1] M.F. Atiyah, K -theory , Mathematics Lecture Note Series, W.A. Benjamin,New York, 1967[2] M.F. Atiyah, F.E.P. Hirzebruch, “Vector bundles and homogeneous spaces”,pp. 7-38 in: Differential geometry , Proc. Sympos. Pure Math. , AmericanMathematical Society, Providence RI, 1961[3] A.-M. Aubert, P.F. Baum, R.J. Plymen, ”The Hecke algebra of a reductive p -adic group: a view from noncommutative geometry”, pp. 1-34 in: Non-commutative geometry and number theory , Aspects of Mathematics
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CLA , 31compactly generated topology, 42continuity, 17, 32, 39convolution, 105coroot, 64coroot lattice, 65Coxeter graph, 62, 66Coxeter group, 62Coxeter system, 62crossed product, 223cuspidal pair, 109cyclic bicomplex, 15cyclic homology, 16 D D ( T ), 141 D σ , 110 d α,(cid:15) , 148 d I , 22 D uv ( I ), 136De Rham homology, 20, 34∆, 84∆ , 114 δ , 114 δ ∅ , 84∆ P , 84∆( P, A ), 114˜ δ γ , 147˜ δ kn , 87density theorem, 37diffeotopy, 34differential forms, 20, 34 E e ρ , 111 e K , 107 e P , 147 e s , 110 EP , 78equivariant Chern character, 46equivariant cohomology, 49equivariant K -theory, 46 equivariant Poincar´e lemma, 53equivariant triangulation, 55 η I , 136 η i , 68, 134Eul, 78Euler characteristic, 78Euler-Poincar´e pairing, 78excision, 18, 33, 39exponent, 116extended quotient, 45 F F (cid:48) , 147 f ( x, q ), 140 f ∗ , 112 F (cid:48) (cid:15) , 154 F , 65 F ∨ m , 66 f p , 165 FA , 31 φ (cid:15) , 156finite type algebra, 23Fr´echet, 49 φ θ,q , 144five lemma, 18 φ W t , 160 F , 91, 120 F , 106Fourier transform, 91, 120Fr´echet algebra, 29 G (cid:104) G (cid:105) , 45 (cid:104) g (cid:105) , 45 G ( φ ), 160 G ( C ∗ r ( R , q )), 160 G ( H ), 78 G K , 50Γ, 143Γ δ , 147Γ rr (Ξ; End( V Ξ )), 91Γ rr (Ξ; L Ξ ), 121gcd( µ ), 205Gelfand topology, 42generalized trace map, 41ndex 239Grothendieck group, 37 G -vector bundle, 45 H H , 70 H ( G ), 107 H ( G ) s , 109 H ( G, K ), 107 H ( G, K ) s , 110 H ( G, K, ρ ), 110 H ( R ), 134 H ( R , I, q ), 74 H ( R ) ∗ , 141 H ( R , q ), 79 H ( W, q ), 63 H ( W P ), 73 H an ( U ), 88 H Gn ( βG ), 127 H me ( U ), 88 H P , 73 H t , 86 H (cid:15) , 148 H M , 113 H P , 84 H U , 72Hecke algebra, 107affine, 68asymptotic, 133Hochschild homology, 16Hochschild-Kostant-Rosenbergtheorem, 20holomorphic functional calculus,29, 140homotopy, 21, 34H-unital, 18strongly, 33 I ı w , 89 I ( k, ω ), 119 I ( n, ω ), 119 I ( P, σ, ν ), 118 I GP ( σ ), 109 I stp , 23 I t , 71 inertial equivalence, 109intertwiner, 90 ı ow,(cid:15) , 148Irr( G ), 106Iwahori subgroup, 107Iwahori-Hecke algebra, 63extended, 64 J J ( P, σ, ν ), 118 J U , 72 J cU , 160 J sU , 160Jac( A ), 22Jacobson topology, 22Jacquet module, 115 Kk , 23 K ( ω, P ), 121 K ω , 119 K , 107 K ( H ), 78 K +0 , 37 K Rep0 ( φ ), 161 K Gj ( βG ), 128 K P , 70 K , 49 K , 36 K -theory, 36 K Ξ , 121 L L , 49 L ( ω, P ), 119 L (Ξ , µ ; End( V Ξ )), 91 L ( G ), 112 L temp , 85 L R , 134 L u , 50, 52label function, 63, 67generic, 85Λ, 73, 118 λ , 79, 112Λ + , 73, 11840 Index λ ( N ), 141Langlands classification, 73, 118Langlands data, 73Langlands dual group, 132length function, 62length-multiplicative, 63 (cid:96) , 62local coefficient system, 49localization, 72locally convex algebra, 28 M M ( ξ ), 123 M (cid:15) , 151 MA , 31Macdonald’s c -function, 84m-algebra, 29manifold, 29Max( A ), 42maximal ideal space, 42metrizable, 28 µ , 91, 205 µ /r , 214 µ P l , 92 µ ∨ , 205 N N , 80, 114 N w , 64, 68nilpotent, 21non-Archimedean local field, 106noncommutative tori, 167non-Hausdorff manifold, 27normalized induction, 109nuclear vector space, 31 ν n , 115 ν ( ξ ), 123 O O (Ξ; End( V Ξ )), 91 O ( Y, Z ), 58 O ( Y, Z ; V ), 58 O , 106Ω, 67Ω (cid:48) , 80 ( ω, E ), 119Ω I , 74 ω ( ξ ), 123orbifold, 48 P P ( ξ ), 93 P ( k, n ), 219 P ( ξ ), 123( P, A ), 113 p δ ( u ), 147 P , 113 p n , 80 P n ( V ), 75 p ( k ) n , 146 p -adic field, 106( P, A, ω, χ ), 120(
P, A ) ≥ ( Q, B ), 113parabolic root subsystem, 65parabolic subalgebra, 63, 69parabolic subgroup, 63parahoric subgroup, 107¯ P , 113periodic cyclic homology, 16periodicity exact sequence, 17perturbations, 165 φ t , 83 π ( g, ξ ), 87, 90 π ( k, ξ ), 87 π ( n, ξ ), 90 π ( P, σ ), 73 π Q ( ξ ), 93 π ( P, σ, t ), 83(˘ π, ˘ V ), 108 π ( ξ ), 84, 120Plancherel measure, 92 P , 106p-pair, 113Prim( A ), 22, 28primitive ideal, 22projective linear map, 154proper root of unity, 132 ψ k , 87Ψ kn ( δ ), 87 ψ n , 87ndex 241 ψ t , 72( P, σ, ν ), 118(
P, W P r, δ, t ), 84 Q Q , 65 q , 63 Q ( A ), 84 Q ∨ , 65 q (cid:15) , 148 q , 82 q P , 69 q α , 113 q i , 63, 134 q P , 69Q-algebra, 29 R R , 65 R ( A ), 174 R ( A ) ∨ , 171 R ( A ) ∨ , 183 R ( GL ), 179 R × R (cid:48) , 65 R ∨ , 65 R P , 65 R pL , 85 R zL , 85 r σ , 73 R − , 65 R +0 , 65 R , 67 R L , 85 R nr , 67 R P , 65, 66 R ∨ P , 65Rad t , 86reduced C ∗ -algebra, 80, 112reduced expression, 62Rep ( C ∗ r ( R , q )), 153Rep( G ), 106Rep( G ) s , 109Rep( H ), 70Rep ( S ( R , q )), 153Rep ρ ( G ), 111 Rep U ( C ∗ r ( R , q )), 153Rep U ( H ( R , q )), 72Rep U ( S ( R , q )), 153representable K -theory, 42representationadmissible, 108anti-tempered, 72contragredient, 108discrete series, 81, 116essentially tempered, 72parabolically induced, 84, 109principal series, 71smooth, 106square-integrable, 116Steinberg, 81supercuspidal, 108tempered, 72, 81, 117trivial, 81residual coset, 85residual point, 85 ρ , 79, 112, 134 ρ , 150 ρ (cid:15) , 148 ρ (cid:15) , 153right regular representation, 225root, 64negative, 65positive, 65simple, 65root datum, 65dual, 65root lattice, 65root system, 65 S S ( G ), 115 S ( G ) s , 115 S ( G, K ), 115 S ( R ), 134 S ( R , Q ), 143 S ( R , q ), 80 s α , 64 S , 66 s ∨ α , 64 S aff , 6642 IndexSchwartz algebra, 81, 115seminorm, 28semisimple algebra, 131sheaf, 25Σ, 49 σ , 114Σ , 114 σ (cid:15) , 148 σ ( µ ), 205 σ n , 144 σ n,q , 144Σ( Q, A ), 113˜ σ (cid:15) , 153simple reflections, 62smooth compact operators, 36smooth map, 107smooth suspension, 39spectrum, 29spectrum preserving, 23weakly, 23stability, 17, 32, 39standard filtration, 23strict inductive limit, 32 T T , 68 t , 65 t ∗ , 65 T L , 85 T n , 167 T P , 70 t P , 70 t P, + , 70 T L , 85 T P , 70 T P,rs , 70 T P,u , 70 T Prs , 70 T P, + rs , 70 T u , 82 T Pu , 70 T w , 63 t x , 66 τ , 79, 112tempered form, 85 tempered function, 117tempered functional, 81 ⊗ , 30 ⊗ , 31 (cid:98) ⊗ , 30tensor productalgebraic, 30inductive, 30injective, 31projective, 30topological, 30over a ring, 31Θ, 23 θ x , 69Tits’s deformation theorem, 131topological algebra, 28topologically pure extension, 33type, 111 U U , 88 U (cid:15) , 148 u γ , 147 u γ,(cid:15) , 154 u g , 49 u I , 136 U t , 88unitization, 16unramified character, 109 V v , 106 V ( N ), 115 V ∞N ( R , q ), 141 V P , 115 V t , 71 V U , 72 V χ , 116 V δ , 84 V Ξ , 84 W W , 66 W ( P, Q ), 87 W P , 73ndex 243 W Q , 93 W σ , 110 W , 66, 114 w , 83 W aff , 66 W I , 74 W P Q , 87weight lattice, 65Weyl group, 66affine, 66 W , 87, 120 X (cid:101) X , 45 X − , 67 x ∗ , 79 X + , 67 X g , 45 X P , 66 X π ( P, A ), 116 X alg , 57 X nr ( H ), 109 X P , 66 X p , 24 X unr ( H ), 109 χ ν , 113Ξ, 84, 114, 120Ξ + , 94Ξ Q , 93Ξ u , 84 Y Y P , 66 Y P , 66 Y p , 24 Z Z , 72 Z ( M ( A )), 23 Z an ( U ), 88 Z me ( U ), 88 Z G ( g ), 45 ζ , 18344 Index amenvatting De afgelopen jaren is mij vaak gevraagd wat ik nou eigenlijk onderzoek. Op dezevraag heb ik inmiddels een voorraadje antwoorden uitgeprobeerd, die in zekere zinallemaal wel correct waren. Niettemin bleek dat sommige antwoorden aanzienlijkmeer begrip en waardering oogsten dan andere. E´en persoon besloot zelfs geheelaf te zien van verdere communicatie nadat ik haar de titel van dit proefschrift hadverteld.Daarom lijkt het me wel een goed idee om in ieder geval de wellicht enigszinscryptische zinsnede
Periodiek cyclische homologie van affiene Hecke algebra’s toete lichten. Dit wordt dan ook niet zozeer een samenvatting van mijn onderzoek, alswel een relatief gezellige wandeling langs de randen van de oneindig dimensionaleruimten waarin ik mij gewoonlijk begeef.Laten we beginnen bij groepen. Met groepen kun je de symmetrie¨en beschrijvenvan uiteenlopende dingen, zoals een voetbal, een velletje papier, een molecuul, eendifferentiaalvergelijking of ruimte-tijd, maar ook van simpele figuren als een lijn,een kubus of een zevenhoek. Een eenvoudige groep, die een rol speelt in dit boek,bestaat uit de zes symmetrie¨en van een gelijkzijdige driehoek.We zien drie spiegelingen (in de stippellijnen) en rotaties over 120 ◦ en over 240 ◦ .De laatste symmetrie is de afbeelding die alles op zijn plek laat. We zien directdat groep binnen de wiskunde iets heel anders betekent dan in het dagelijks leven.In abstracto is het een verzameling waarvan je de elementen kunt samenstellen,waarbij aan bepaalde voorwaarden voldaan moet zijn.De bovenstaande groep heeft allerlei bijzondere eigenschappen. Bijvoorbeeld,als je een willekeurige lijn neemt die door het middelpunt van de driehoek gaat,24546 Samenvattingen je past daarop een symmetrie van de driehoek toe, dan krijg je weer een lijndie door dat middelpunt gaat. Men zegt daarom wel dat deze groep bestaat uitlineaire afbeeldingen.Beschouw het rooster dat is opgebouwd uit gelijkzijdige driehoeken. (Zie dekaft voor een artistiekere impressie van dit rooster, waarvoor mijn dank uitgaatnaar Bill Wenger.)De symmetriegroep van dit (oneindig grote) figuur bevat oneindig veel elementen,onder andere spiegelingen en rotaties, maar ook verschuivingen. Stel dat je eenpunt in dit rooster kiest, en een lijn door dat punt. Als je zomaar een symmetrievan het rooster toepast krijg je zeker weer een lijn, maar er is geen garantie dat dienog steeds door het gekozen punt gaat. Daarom noemt men dit soort symmetrie¨engeen lineaire afbeelingen, maar affiene afbeeldingen.Met deze hele opzet kunnen we iets doen waar wiskundigen dol op zijn, wekunnen het zaakje generaliseren. In dat geval vervangen we de driehoek door eeningewikkelder kristal, en het driehoekige rooster door een rooster van hogere di-mensie. De symmetriegroep van het kristal is een (eindige) Weyl groep, en desymmetriegroep van het rooster heet een affiene Weyl groep.Als je een groep goed wilt begrijpen is het van groot belang om zijn zogehetenrepresentaties te kennen. Dit illustreren we aan de hand van ander voorbeeld, decirkel. Enerzijds kan deze worden beschouwd als de groep van rotaties om zijneigen middelpunt. Anderszijds kunnen we de cirkel opvatten als een snaar, en dankan hij trillen. De representaties van de cirkelgroep corresponderen precies met detrillingen van de cirkelvormige snaar, waarbij we ´e´en specifiek punt P vasthouden. P P PP P
Er is een grondtoon, waarvan de golflengte exact de lengte van de snaar is. Deboventonen hebben een golflengte die een geheel aantal keren in de snaar past.amenvatting 247Elke trilling is te maken als een geschikte combinatie van dergelijke harmonischetrillingen.In wiskundig jargon betekent dit dat de harmonische trillingen corresponderenmet de irreducibele representaties. Irreducibele representaties zijn een soort bouw-steentjes waarmee je elke representatie kunt maken. Ze zijn zelf niet verder op tedelen.Grof gezegd is een algebra een verzameling waarbinnen je kunt optellen envermenigvuldigen. Zo vormen de gehele getallen een algebra. Een wat lastigervoorbeeld zijn de 2 × M ( R ) := (cid:26)(cid:18) x x x x (cid:19) : x , x , x , x ∈ R (cid:27) Dit soort matrices kan je co¨ordinaatsgewijs optellen: (cid:18) π (cid:19) + (cid:18) / − / − − (cid:19) = (cid:18) / − / π − (cid:19) Een matrix is op te vatten als een lineaire afbeelding. Als we punten in het vlakschrijven als vectoren (cid:18) xy (cid:19) , dan stuurt een matrix dus een vector naar een anderevector: (cid:18) / − / − − (cid:19) (cid:18) a (cid:19) = (cid:18) a/ − a (cid:19) (cid:18) / − / − − (cid:19) (cid:18) b (cid:19) = (cid:18) − b/ − b (cid:19) De standaardmanier om matrices te vermenigvuldigen correspondeert met hetsamenstellen van afbeeldingen, bijvoorbeeld (cid:18) (cid:19) · (cid:18) (cid:19) = (cid:18) (cid:19) (cid:18) (cid:19) · (cid:18) (cid:19) = (cid:18) (cid:19) Al met al maakt dit M ( R ) tot een algebra over de re¨ele getallen R . Merk op datbepaalde gebruikelijke rekenregels voor getallen niet meer gelden voor matrices.We zijn gewend dat 7 · · · · x · y = y · x voor alle re¨ele getallen x en y . Niettemin zien we hierbovendat er matrices A en B bestaan zodat A · B (cid:54) = B · A Men zegt dan dat A en B niet commuteren en dat de algebra M ( R ) niet commu-tatief is.Wat is nu het verband tussen algebra’s en groepen? Vanuit een groep kunnenwe een algebra construeren die alle eigenschappen van de groep reflecteert. Zo48 Samenvattinghebben algebra’s ook representaties, en de representaties van een groep komenovereen met de representaties van de bijbehorende groepsalgebra.Een aanzet tot de algebra’s in de titel van dit boek werd gegeven door deDuitse wiskundige Erich Hecke, die leefde van 1887 tot 1947. Hecke hield zichvooral bezig met getaltheorie, bijvoorbeeld met p -adische getallen. Hier is p eenpriemgetal, bijvoorbeeld 5. De verzameling 5-adische getallen geven we aan met Q . Een typisch 5-adisch getal ziet eruit als een decimale expansie in de verkeerderichting: x = · · · . ∈ Q Omdat p = 5 komen alleen de symbolen 0, 1, 2, 3 en 4 in deze schrijfwijze van x voor. We dienen x te interpreteren als x = 3 · − + 1 · − + 1 · − + 4 · + 4 · + 0 · + 2 · + · · · Als y een ander 5-adisch getal is, dan zijn x + y en x · y zonder al te veel problemente bepalen met behulp van de regeltjes a n + b n = ( a + b )5 n a n · c m = ( a · c )5 n + m voor gehele getallen a, b, c, n en m . Om de co¨efficient van x · y bij 5 n uit te rekenenhebben we slechts kleine stukjes van de expansies van x en y nodig. Zelfs delen ismogelijk met p -adische getallen, omdat p een priemgetal is.Met p -adische getallen kunnen we weer groepen bouwen. Een simpel voorbeeldvan zo’n p -adische groep is (cid:26)(cid:18) a bc d (cid:19) : a, b, c, d ∈ Q , a · d − b · c = 1 (cid:27) De samenstelling in deze groep is de gebruikelijke matrixvermeningvuldiging, maardan uitgevoerd met 5-adische getallen. Dergelijke p -adische groepen spelen eenbelangrijke rol in verschillende gebieden van de wiskunde. Men zou graag alleirreducibele representaties van zo’n groep klassificeren, maar dat is erg lastig. Hetblijkt dat een belangrijk deel van de representatietheorie van een p -adische groepvalt uit te drukken met een zekere algebra. Zo een algebra is een generalisatie vaneen type algebra’s dat Hecke indertijd vanuit een iets andere hoek heeft gedefinieerden bestudeerd, vandaar dat ze onder de naam Hecke algebra’s door het leven gaan.Hecke algebra’s zijn er in soorten en maten. Ik ben vooral ge¨ınteresseerd inHecke algebra’s die sterk gerelateerd zijn aan Weyl groepen. In feite zijn dezeHecke algebra’s te beschouwen als deformaties van Weyl groepen. De groepsal-gebra die bij een Weyl groep hoort heeft zoals zoals gezegd grotendeels dezelfdeeigenschappen als de groep zelf. Als je die groepsalgebra op een geschikte maniervervormt krijg je een Hecke algebra.Doe je dit met een eindige Weyl groep, dan krijg je een Hecke algebra vaneindige dimensie. Hoewel eindig dimensionaal in eerste instantie nog niet zo een-voudig klinkt, is het vast makkelijker dan oneindig dimensionaal . In ieder gevalbegrijpt men eindig dimensionale Hecke algebra’s heel goed.amenvatting 249Echter, als je een affiene Weyl groep deformeert krijg je een affiene Hecke alge-bra, en die heeft oneindige dimensie. Affiene Hecke algebra’s zijn veel ingewikkelderdan Hecke algebra’s van eindige dimensie. Toch is dat niet zo’n ramp. Men ge-bruikt affiene Hecke algebra’s onder andere om tot een beter begrip te komen vande gecompliceerde representatietheorie van p -adische groepen, en daar zou weinigvan te verwachten zijn als ze te eenvoudig waren. Het blijkt dat affiene Heckealgebra’s aan de ene kant diepzinnig genoeg zijn om tot nieuwe inzichten te leiden,en aan de andere kant makkelijk genoeg om er prettig mee te kunnen werken.Dus een affiene Hecke algebra heeft precies de goede moeilijkheid om hem tot eeninteressant studieobject te maken.Zonet zagen we de analogie tussen trillingen van een snaar en representatiesvan een groep. Als we dit uitbreiden correspondeert een algebra niet meer met ´e´ensnaar, maar met een snaarinstrument, bijvoorbeeld een piano. Een representatievan die algebra wordt dan een toon die je met die piano kunt voortbrengen. Opdeze manier kunnen we een irreducibele representatie identificeren met een zuiveretoon van de piano.Het uiteindelijke doel van mijn promotieonderzoek was om alle irreducibelerepresentaties van een algemene affiene Hecke algebra te bepalen. Het ligt voorde hand om ze eerst maar eens te tellen. Helaas laten ze zich niet zo gemakkelijktellen, want het zijn er oneindig veel. Net zo heeft het weinig zin om alle zuiveretonen van een piano te tellen, want dat zijn er ook oneindig veel. Een beter ideeis daarom om alle grondtonen van de snaren van de piano te tellen, dat vertelt jebijvoorbeeld al hoeveel snaren je piano heeft.In de context van algebra’s ligt dat wat subtieler, daar heet de geschikte manierom grondtonen te tellen periodiek cyclische homologie . ”Homologie” komt uit hetGrieks en betekent zoveel als ”studie van gelijkheid”. Dat gaat ongeveer als volgt.Stel dat je twee objecten, bijvoorbeeld twee algebra’s, wilt vergelijken. Kies eengeschikte methode (een homologietheorie) om aan een algebra iets relatief een-voudigs toe te kennen, bijvoorbeeld een simpel type groep, een rijtje getallen ofzelfs een rijtje groepen. Dat heet dan de homologie van de algebra. Als je tweealgebra’s in essentie hetzelfde zijn zullen ze dezelfde homologie hebben. Daarente-gen, als ze verschillende homologie hebben dan zijn de algebra’s niet hetzelfde, enook niet ongeveer.Op de kaft van dit boek zie je duidelijk periodieke en cyclische verschijnselen.Dat is geen toeval, maar de etymologische achtergrond van de term periodiekcyclische homologie is anders. De periodiek cyclische homologie van een algebra A is een rijtje groepen: . . . , HP − ( A ) , HP − ( A ) , HP ( A ) , HP ( A ) , HP ( A ) , HP ( A ) , . . . Dit is periodiek in de zin dat voor elk geheel getal n geldt HP n ( A ) = HP n +2 ( A )Het cyclische zit wat dieper verstopt, dat heeft te maken met hoe de groepen50 Samenvatting HP n ( A ) expliciet geconstrueerd worden. Stel dat we zeven hokjes hebben, dieallemaal gevuld zijn met een letter:f b d c r t zNu schuiven we elke letter ´e´en hokje naar rechts, en de meest rechtse letter stoppenwe in het vrijgekomen linker hokje:z f b d c r tDit kunnen we nog wat suggestiever tekenen: c dbfzt r c d bfztr Nu is het wel duidelijk waarom dit een cyclische permutatie heet. Zulke permu-taties worden gebruikt in de definitie van periodiek cyclische homologie.Het is nog niet zo eenvoudig om de periodiek cyclische homologie van een affieneHecke algebra ook daadwerkelijk uit te rekenen. Daartoe grijpen we terug op deaffiene Weyl groep waarvan het een deformatie is. Als we weer denken aan piano’sen snaren betekent deze deformatie dat we wat gaan sleutelen aan de snaren: watlanger of iets korter, een stukje dichter bij elkaar. Hoewel het in muzikaal opzichtbarbaars is zouden we zelfs sommige snaren aan elkaar vast kunnen knopen. Zo’ndeformatie kan er schematisch uitzien alsMen vermoedt dat de representatietheorie van een affiene Weyl groep nietessentieel verandert onder deze deformaties. Dit vermoeden wordt ondersteunddoor diepzinnige stellingen die zeggen dat het in bepaalde belangrijke gevallenklopt. Het is een belangrijk vermoeden, want hiermee kan men representaties vaneen affiene Hecke algebra herleiden tot representaties van een affiene Weyl groep,en die zijn allemaal al lang bekend.In dit proefschrift heb ik bewezen dat dit vermoeden equivalent is met eenogenschijnlijk zwakkere uitspraak, namelijk dat de periodiek cyclische homologievan de groepsalgebra van een affiene Weyl groep niet verandert als je die algebravervormt tot een affiene Hecke algebra. Grof gezegd betekent de sterke versievan dit vermoeden dat je met elk van de boven getekende pianootjes evenveelamenvatting 251verschillende tonen kan voortbrengen. De zwakke versie zegt zo ongeveer dat aldie piano’s evenveel grondtonen hebben.Verder worden in dit boek onder andere een aantal nieuwe technieken ge¨ıntro-duceerd om de periodiek cyclische homologie van een redelijk algemeen type al-gebra uit te rekenen. Mede daardoor kunnen de bovenstaande vermoedens nubewezen worden in veel nieuwe gevallen.52 Samenvatting urriculum vitae
De auteur werd geboren op 5 februari 1979 in Amsterdam. Zijn middelbareschooltijd bracht hij door aan het Montessori Lyceum Amsterdam, alwaar hijin juni 1997 voor zijn VWO-examen slaagde.In september 1997 begon hij als student aan de Universiteit van Amsterdam.In 1998 met behaalde hij cum laude een dubbele propedeuse wiskunde en natuur-kunde, waarop hij besloot zich verder te concentreren op de wiskunde. Hij schreefzijn scriptie, getiteld