AAFFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS
Maarten Solleveld
IMAPP, Radboud Universiteit NijmegenHeyendaalseweg 135, 6525AJ Nijmegen, the Netherlandsemail: [email protected]
Abstract.
This is a survey paper about affine Hecke algebras. We start fromscratch and discuss some algebraic aspects of their representation theory, referringto the literature for proofs. We aim in particular at the classification of irreduciblerepresentations.Only at the end we establish a new result: a natural bijection between theset of irreducible representations of an affine Hecke algebra with parameters in R ≥ , and the set of irreducible representations of the affine Weyl group underlyingthe algebra. This can be regarded as a generalized Springer correspondence withaffine Hecke algebras. Contents
Introduction 21. Definitions and first properties 41.1. Finite dimensional Hecke algebras 41.2. Iwahori–Hecke algebras 61.3. Affine Hecke algebras 71.4. Presentations of affine Hecke algebras 101.5. Graded Hecke algebras 122. Irreducible representations in special cases 152.1. Affine Hecke algebras with q = 1 152.2. Graded Hecke algebras with k = 0 162.3. Iwahori–Hecke algebras of type (cid:102) A GL n Date : September 8, 2020.2010
Mathematics Subject Classification. a r X i v : . [ m a t h . R T ] S e p AFFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS
4. Classification of irreducible representations 404.1. Analytic R-groups 414.2. Residual cosets 445. Geometric methods 485.1. Equivariant K-theory 485.2. Equivariant homology 525.3. Affine Hecke algebras from cuspidal local systems 576. Comparison between different q -parameters 606.1. W -types of irreducible tempered representations 606.2. A generalized Springer correspondence with affine Hecke algebras 67References 73 Introduction
Affine Hecke algebras typically arise in two ways. Firstly, they are deformationsof the group algebra of a Coxeter system (
W, S ) of affine type. Namely, keep thebraid relations of W , but replace every quadratic relation s = 1 ( s ∈ S ) by(0.1) ( s − q s )( s + 1) = 0 , where q s is a parameter in some field (usually we take the field C ). That gives riseto an associative algebra H ( W, q ).Secondly, affine Hecke algebras occur in the representation theory of reductivegroups G over p -adic fields. They can be isomorphic to the algebra of G -endo-morphisms of a suitable G -representation. The classical example is the convolutionalgebra of compactly supported functions on G that are bi-invariant with respect toan Iwahori subgroup. In this way the representation theory of affine Hecke algebrasis related to that of reductive p -adic groups. The interpretation of (affine) Heckealgebras as deformations of group algebras links them with quantum groups, knottheory and noncommutative geometry. Further, their relation with reductive groupsmakes them highly relevant in the representation theory of such groups and in thelocal Langlands program.Both views of affine Hecke algebras build upon simpler objects: the Hecke alge-bras of finite Weyl groups. On the one hand such a finite dimensional Hecke algebrais a deformation of a group algebra, again with relations (0.1). On the other hand itappears naturally in the representation theory of reductive groups over finite fields.However, there is a crucial difference between the finite and affine cases: most as-pects of finite dimensional Hecke algebras are easily described in terms of a Coxetergroup, but that is far from true for affine Hecke algebras.Of course there exists an extensicve body of literature on affine Hecke algebrasand their representations, see the references to this paper for a part of it. The theoryis in a good state, and (in our opinion) most of the important questions that onecan ask about affine Hecke algebras have been answered.Unfortunately, the accessibility of this literature is rather limited. A first difficultyis that several slightly different algebras are involved, with several presentations, andsometimes their connection is not clear. Further, a plethora of techniques has beenapplied to affine Hecke algebras: algebraic, analytic, geometric or combinatoric to FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 3 various degrees. Finally, to the best of our knowledge no textbook treats affineHecke algebras any further than their presentations.With this survey paper we intend to fill a part of that gap. Our aim is an in-troduction to affine Hecke algebras and their representations, of which the largerpart is readable for mathematicians without previous experience with the subject.To ease the presentation, we will hardly prove anything, and we will provide manyexamples. As a consequence, our treatment is almost entirely algebraic - the deep an-alytic or geometric arguments behind some important results are largely suppressed.Let us discuss the contents in a nutshell. In the first section we work out the var-ious presentations of (affine) Hecke algebras over C , and we compare them. Section2 consists of explicit examples: we look at the most frequent affine Hecke algebrasand provide an overview of their irreducible representations.The third section is the core of the paper, here we build up the abstract repre-sentation theory of an affine Hecke algebra H . This is done in the spirit of Harish-Chandra’s analytic approach to representations of reductive groups: we put theemphasis on parabolic induction, the discrete series and the large commutative sub-algebra of H . To make this work well, we need to assume that the parameters q s of H lie in R > . With these techniques one can divide the set of irreducible represen-tations Irr( H ) into L-packets, like in the local Langlands program. To achieve more,it pays off to reduce from affine Hecke algebras to a simpler kind of algebras calledgraded Hecke algebras. This is like reducing questions about a Lie group to its Liealgebra.More advanced techniques to classify Irr( H ) are discussed in Section 4. In princi-ple this achieves a complete classification, but in practice some computations remainto be done in examples.In Section 5 we report on algebro-geometric approaches to affine Hecke algebrasand graded Hecke algebras, largely due to Lusztig. In many cases, these methodsyield beautiful constructions and parametrizations of all irreducible representations.Although we treat the involved (co)homology theories as a black box, we do provide acomparison between these constructions and the setup inspired by Harish-Chandra.The final section of the paper is quite different from the rest, here we do actuallyprove some new results. The topic is the relation between an affine Hecke algebra H with parameters q s ∈ R ≥ and its version with parameters q s = 1. The latter algebrais of the form C [ X (cid:111) W ], where X is a lattice and W is the (finite) Weyl group of aroot system R in X . We note that X (cid:111) W contains the affine Weyl/Coxeter group Z R (cid:111) W . There is a natural way to regard any finite dimensional H -representation π as a representation of C [ W ], we call that the W -type of π . Theorem A. (see Theorem 6.12)Let H be an affine Hecke algebra with parameters in R ≥ (and a mild condition when R has components of type F ). There exists a natural bijection ζ (cid:48)H : Irr( H ) −→ Irr( X (cid:111) W ) such that the restriction of ζ (cid:48)H ( π ) to W is always a constituent of the W -type of π . It is interesting to restrict Theorem A to irreducible X (cid:111) W -representations onwhich X acts trivially (those can clearly be identified with irreducible W -represen-tations). The inverse image of these under ζ (cid:48)H is the set Irr ( H ) of irreducible AFFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS tempered H -representations “with real central character”, see Paragraph 6.1. When H is of geometric origin, Irr ( H ) is naturally parametrized by data as in Lusztig’sgeneralization of the Springer correspondence with intersection homology, see Para-graph 5.2. In that case the bijection ζ (cid:48)H : Irr ( H ) → Irr( W )becomes an instance of the generalized Springer correspondence. Moreover, for suchgeometric affine Hecke algebras the whole of Irr( H ) admits a natural parametrizationin terms of data that are variations on Kazhdan–Lusztig parameters, see Paragraph5.3. Via this parametrization, ζ (cid:48)H becomes a generalized Springer correspondencefor the (extended) affine Weyl group X (cid:111) W .With that in mind we can regard Theorem A, for any eligible H , as a “generaliza-tion of the Springer correspondence with affine Hecke algebras”. This applies bothto the finite Weyl group W and the (extended) affine Weyl group X (cid:111) W .Of course the selection of topics in any survey is to a considerable extent the tasteof the author. To preserve a reasonable size, we felt forced to omit many interestingaspects of affine Hecke algebras: the Kazhdan–Lusztig basis [KaLu1], asymptoticHecke algebras [Lus7], unitary representations [BaMo1, Ciu], the Schwartz and C ∗ -completions of affine Hecke algebras [Opd2, DeOp1], homological algebra [OpSo1,Sol5], formal degrees of representations [OpSo2, CKK], spectral transfer morphisms[Opd3, Opd4] and so on. We apologize for these and other omissions and refer thereader to the literature.1. Definitions and first properties
Finite dimensional Hecke algebras.
Let (
W, S ) be a finite Coxeter system – so W is a finite group generated by a set S of elements of order 2. Moreover, W has a presentation W = (cid:104) S | ( ss (cid:48) ) m ( s,s (cid:48) ) = e ∀ s, s (cid:48) ∈ S (cid:105) , where m ( s, s (cid:48) ) ∈ Z ≥ is the order of ss (cid:48) in W . The equalities s = e ( s ∈ S ) arecalled the quadratic relations, while ( ss (cid:48) ) m ( s,s (cid:48) ) = e , or equivalently s s (cid:48) s s (cid:48) · · · (cid:124) (cid:123)(cid:122) (cid:125) m ( s,s (cid:48) ) terms = s (cid:48) s s (cid:48) s · · · (cid:124) (cid:123)(cid:122) (cid:125) m ( s,s (cid:48) ) terms is known as a braid relation. Examples to keep in mind are • W = S n , S = { (12) , (23) , . . . , ( n − n ) } – type A n − ; • W = S n (cid:110) {± } n , S = { (12) , (23) , . . . , ( n − n ) , (id , (1 , . . . , , − } – type B n or C n .In the group algebra C [ W ] the quadratic relations are equivalent with(1.1) ( s + 1)( s −
1) = 0 s ∈ S. Now we choose, for every s ∈ S , a complex number q s , such that(1.2) q s = q s (cid:48) if s and s (cid:48) are conjugate in W. Let q : S → C be the function s (cid:55)→ q s . We define a new C -algebra H ( W, q ) whichhas a vector space basis { T w : w ∈ W } . Here T e is the unit element and there arequadratic relations ( T s + 1)( T s − q s ) = 0 s ∈ S FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 5 and braid relations(1.3) T s T s (cid:48) T s · · · (cid:124) (cid:123)(cid:122) (cid:125) m ( s,s (cid:48) ) terms = T s (cid:48) T s T s (cid:48) · · · (cid:124) (cid:123)(cid:122) (cid:125) m ( s,s (cid:48) ) terms . Equivalent versions of these quadratic relations are(1.4) T s = ( q s − T s + q s T e and T − s = q − s T s + ( q − s − T e (the latter only when q s (cid:54) = 0). For q = 1, the relations (1.3) become the definingrelations of the Coxeter system ( W, S ), so H ( W,
1) = C [ W ]. We say that H ( W, q )has equal parameters if q s = q s (cid:48) for all s, s (cid:48) ∈ S .The condition (1.2) is necessary and sufficient for the existence of an associativeunital algebra H ( W, q ) with these properties [Hum, § H ( W, q ) has the structure of a symmetric algebra: it carries a trace τ ( T w ) = δ w,e , an involution T ∗ w = T w − and a bilinear form ( x, y ) = τ ( x ∗ y ). Thesealso give rise to interesting properties [GePf], which however fall outside the scopeof this survey.Without the trace τ , the representation theory of finite dimensional Hecke algebrasis quite easy. To explain this, we consider a more general situation.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(1.5) T g · T h = (cid:88) w ∈ G a g,h,w T w 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(1.6) T g · q T h = (cid:88) w ∈ G a g,h,w ( q ) T w 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’ deformation theorem, see [Car, p. 357 - 359] or [Iwa2, Appendix]. Theorem 1.1.
There exists a polynomial P ∈ k such that the following are equiva-lent : • P ( q ) (cid:54) = 0 , • H ( G, q ) is semisimple, • H ( G, q ) ∼ = C [ G ] . In other words: when H ( G, q ) is semisimple it is isomorphic to C [ G ], and otherwisethe algebra H ( W, q ) has nilpotent ideals and looks very different from C [ G ]. The AFFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS simplest case where the latter occurs is already G = { e, s } . Namely, for q = − H ( G, −
1) = C [ T s + 1] / ( T s + 1) . Let us discuss how it works out for an arbitrary finite Coxeter system (
W, S ) and aparameter function q as before. For an element w ∈ W with a reduced expression s s · · · s r we put(1.8) q ( w ) = q s q s · · · q s r . This is well-defined by the presentation of W and condition (1.2). We want to knowunder which conditions the algebra H ( W, q ) is semisimple. For such groups thepolynomials P ( q ) of Theorem 1.1 have been determined explicitly. If we are in theequal label case q ( s ) = q ∀ s ∈ S then we may take(1.9) P ( q ) = q (cid:88) w ∈ W q (cid:96) ( w ) except that we must omit the factor q if W is of type ( A ) n , see [GyUn]. Moregenerally, suppose that if ( W, S ) is irreducible and S consists of two conjugacyclasses, with parameters q and q . Gyoja [Gyo, p. 569] showed that in most ofthese cases we may take(1.10) P ( q , q ) = q | W | q W ( q , q ) W ( q − , q ) W ( q , q ) = (cid:80) w ∈ W q ( w )So generically there is an isomorphism(1.11) H ( W, q ) ∼ = C [ W ] . In particular, for almost all q the representation theory of H ( W, q ) is just that of W (over C ).When q q (cid:54) = 0 and W ( q , q ) = 0, the subgroup of C × generated by q and q contains a root of unity different from 1. Hence the non-semisimple algebras H ( W, q )are those with (at least) one q s equal to 0 and some for which q involves nontriv-ial roots of unity. Although these algebras can have an interesting combinatorialstructure, their behaviour is quite different from that of affine Hecke algebras.1.2. Iwahori–Hecke algebras.
Let (
W, S ) be any Coxeter system. As in the previous paragraph, we can assigncomplex numbers q s to the elements of S . When (1.2) is fulfilled, we can constructan algebra H ( W, q ) exactly as before. For q = 1 this is a group algebra of W , andhence, for q (cid:54) = 1, H ( W, q ) can be regarded as some deformation of C [ W ]. However,the situation is much more complicated than when W is finite. Tits’ deformationtheorem does not work here, and in general many different q ’s can lead to mutuallynon-isomorphic algebras [Yan].To get some grip on the situation, we restrict our scope from general Coxetergroups to affine Weyl groups. By that we mean Coxeter systems ( W aff , S aff ) such thatevery irreducible component S i of S aff generates a Coxeter group W i of affine type.The affineness condition is equivalent to: the Cartan matrix of ( W i , S i ) is positivesemidefinite but not positive definite [Hum, § § § A n , ˜ B n , ˜ C n , ˜ D n , ˜ E , ˜ E , ˜ E , ˜ F , ˜ G .In the simplest case ˜ A , W aff is an infinite dihedral group, freely generated by twoelements of order 2. FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 7
Definition 1.2.
An Iwahori–Hecke algebra is an algebra of the form H ( W, q ), where(
W, S ) is any Coxeter system. We say that H ( W, q ) is of affine type if (
W, S ) is anaffine Weyl group.Iwahori–Hecke algebras of affine type where discovered first by Matsumoto andIwahori [Mat1, IwMa, Iwa2], in the context of reductive p -adic groups. For instance,let G be a split, simply connected, semisimple group over Q p and let I be an Iwahorisubgroup of G . Then the convolution algebra C c ( I \ G/I ) is isomorphic to H ( W aff , q ),where ( W aff , S aff ) is derived from G and q s = p for all s ∈ S . This is called theIwahori-spherical Hecke algebra of G , because its modules classify G -representationswith I -fixed vectors.The basic structure of an affine Weyl group W aff is described in [Hum, § § VI.2]. The set of elements whose conjugacy class is finite forms a finiteindex normal subgroup of W aff , isomorphic to a lattice. That lattice is spanned byan integral root system R , and W aff is the semidirect product of Z R and the Weylgroup of R . In particular H ( W aff , q ) contains H ( W ( R ) , q ) as a subalgebra. However,the embedding of W ( R ) in W aff is in general not unique.1.3. Affine Hecke algebras.
Affine Hecke algebras generalize Iwahori–Hecke algebras of affine type. Insteadof affine Weyl groups, we allow more general groups which are semidirect productsof lattices and finite Weyl groups. The best way to do the bookkeeping is with rootdata, for which our standard references are [Bou, Hum].Consider a quadruple R = ( X, R, Y, R ∨ ), where • X and Y are lattices of finite rank, with a perfect pairing (cid:104)· , ·(cid:105) : X × Y → Z , • R is a root system in X , • R ∨ ⊂ Y is the dual root system, and a bijection R → R ∨ , α (cid:55)→ α ∨ with (cid:104) α, α ∨ (cid:105) = 2 is given, • for every α ∈ R , the reflection s α : X → X, s α ( x ) = x − (cid:104) x, α ∨ (cid:105) α stabilizes R , • for every α ∨ ∈ R ∨ , the reflection s ∨ α : Y → Y, s ∨ α ( y ) = y − (cid:104) α, y (cid:105) α ∨ stabilizes R ∨ .If all these conditions are met, we call R are root datum. It comes with a finiteWeyl group W = W ( R ) and an infinite group W ( R ) = X (cid:111) W ( R ), the extendedaffine Weyl group of R . Often we will add a base of R to R , and speak of a basedroot datum. Example 1.3.
Take X = Y = Z , R = {± } and R ∨ = {± } . Then(1.12) W ( R ) = Z (cid:111) S , an infinite dihedral group.We stress that we do not require R to span the vector space X ⊗ Z R . We say that R is semisimple if R does span X ⊗ Z R . For non-semisimple root data R and R ∨ may even be empty. For instance, root datum R = ( Z n , ∅ , Z n , ∅ ) has infinite group W ( R ) = Z n , but no reflections. AFFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS
More examples of root data (actually all) come from reductive groups, see [Spr].Suppose that G is a reductive algebraic group, T is a maximal torus in G and R ( G , T )is the associated root system. Denote the character lattice of T by X ∗ ( T ) and itscocharacter lattice by X ∗ ( T ). Then(1.13) R ( G , T ) := (cid:0) X ∗ ( T ) , R ( G , T ) , X ∗ ( T ) , R ( G , T ) ∨ (cid:1) . is a root datum. We note that R ( G , T ) is semisimple if and only if G is semisimple.The group W ( R ) = X (cid:111) W acts naturally on the vector space X ⊗ Z R , X bytranslations and W by linear extension of its action on X . The collection of hyper-planes H α,n = { x ∈ X ⊗ Z R : α ∨ ( x ) = n } α ∈ R, n ∈ Z is W ( R )-stable and divides X ⊗ Z R in open subsets called alcoves. Let W aff be thesubgroup of W ( R ) generated by the (affine) reflections in the hyperplanes H α,n .This is an affine Weyl group, and X ⊗ Z R with this hyperplane arrangement is itsCoxeter complex.To construct Hecke algebras from root data, we need to specify a set of Coxetergenerators of W aff . In this setting they will be affine reflections. Let ∆ be a baseof R . As is well-known, it yields a set of simple reflections S = { s α : α ∈ ∆ } , and( W, S ) is a finite Coxeter system. (For a root datum of the form R ( G , T ) as in (1.12),the choice of ∆ is equivalent to the choice of a Borel subgroup B ⊂ G containing T .)The base ∆ determines a “fundamental alcove” A in X ⊗ Z R , namely the uniquealcove contained in the positive Weyl chamber (with respect to ∆), such that 0 ∈ A .The reflections in those walls of A that contain 0 constitute precisely S . The set S aff of (affine) reflections with respect to all walls of A forms the required collectionof Coxeter generators of W aff . Example 1.4.
A part of the hyperplane arrangement for an affine Weyl group oftype ˜ A , with A = A , ∆ = { α, β } and S aff = { s α , s β , s γ } . s A s As A HHHH H H α,0 α,1α,−1 β,0β,1α+β,0α+β,1βα γ s s A β αβ H Α γ We can make S aff more explicit. Let R ∨ max be the set of maximal elements of R ∨ ,with respect to the base ∆ ∨ . It contains one element for every irreducible componentof R ∨ . For α ∨ ∈ R ∨ max , define s (cid:48) α : X → X, s (cid:48) α ( x ) = x + α − (cid:104) x, α ∨ (cid:105) α. FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 9
This is the reflection of X ⊗ Z R in the hyperplane H α, , a wall of A . Then S aff = S ∪ { s (cid:48) α : α ∨ ∈ R ∨ max } . We denote the based root datum (
X, R, Y, R ∨ , ∆) also by R . Thus the sets S, S aff and the subgroup W aff ⊂ W ( R ) are determined by R . Example 1.5.
A couple of important instances, coming from the reductive groups
P GL and GL n : • X = Y = Z , R = {± } , R ∨ = {± } , ∆ = { α = 2 } . Here S aff = { s α , s (cid:48) α : x (cid:55)→ − x } and W aff = 2 Z (cid:111) S , a proper subgroup of W ( R ) = Z (cid:111) S . This is thesame W ( R ) as in (1.12), but with a different set of simple affine reflections. • X = Y = Z n , R = R ∨ = A n − = { e i − e j : 1 ≤ i, j ≤ n, i (cid:54) = j } , ∆ = { e i − e i +1 : i = 1 , . . . , n − } . In this case S aff = { s i = s e i − e i +1 : i = 1 , . . . , n − } ∪ { s : x (cid:55)→ x + (1 − (cid:104) x, e − e n (cid:105) )( e − e n ) } W aff = { ( x , . . . , x n ) ∈ Z n : x + · · · + x n = 0 } (cid:111) S n (1.14)Like Iwahori–Hecke algebras, affine Hecke algebras involve q -parameters. Fix q ∈ R > and let λ, λ ∗ : R → C be functions such that • if α, β ∈ R are W -associate, then λ ( α ) = λ ( β ) and λ ∗ ( α ) = λ ∗ ( β ), • if α ∨ / ∈ Y , then λ ∗ ( α ) = λ ( α ).We note that α ∨ ∈ Y is only posssible for short roots α in a type B component of R . For α ∈ R we write(1.15) q s α = q λ ( α ) and (if α ∨ ∈ R ∨ max ) q s (cid:48) α = q λ ∗ ( α ) . Recall that H ( W, q ) is the Iwahori–Hecke algebra of W = W ( R ) and let { θ x : x ∈ X } be the standard basis of C [ X ]. Definition 1.6.
The affine Hecke algebra H ( R , λ, λ ∗ , q ) is the vector space C [ X ] ⊗ C H ( W, q ) with the multiplication rules: • C [ X ] and H ( W, q ) are embedded as subalgebras, • for α ∈ ∆ and x ∈ X : θ x T s α = T s α θ s α ( x ) = (cid:16) ( q λ ( α ) − θ − α (cid:0) q ( λ ( α )+ λ ∗ ( α )) / − q ( λ ( α ) − λ ∗ ( α )) / (cid:1)(cid:17) θ x − θ s α ( x ) θ − θ − α . When α ∨ / ∈ Y , the cross relation simplifies to θ x T s α = T s α θ s α ( x ) = ( q λ ( α ) − θ x − θ s α ( x ) )( θ − θ − α ) − . Notice that here the right hand side lies in C [ X ] because θ x − θ s α ( x ) = θ x − θ x −(cid:104) x,α ∨ (cid:105) α = θ x ( θ − θ −(cid:104) x,α ∨ (cid:105) α )is divisible by θ − θ − α . It follows from [Lus3, §
3] that H ( R , λ, λ ∗ , q ) is really anassociative algebra with unit element θ ⊗ T e , and that the multiplication map(1.16) H ( W, q ) ⊗ C C [ X ] → H ( R , λ, λ ∗ , q ) h ⊗ f (cid:55)→ h · f is bijective. We say that H ( R , λ, λ ∗ , q ) has equal parameters if λ ( α ) = λ ( β ) = λ ∗ ( α ) = λ ∗ ( β ) for all α, β ∈ R. When λ = λ ∗ = 1, we omit them from the notation and write simply H ( R , q ).In that setting we will often allow q to be any element of C × . We note that for λ = λ ∗ = 0 or q = 1 we recover the group algebra C [ X (cid:111) W ]. In particular, the onlyaffine Hecke algebra associated to ( X, ∅ , Y, ∅ ) is C [ X ].Affine Hecke algebras appear foremostly in the representation theory of reductive p -adic groups, see [BuKu1, Mor, Roc, ABPS2, Sol8]. They also have strong ties toorthogonal polynomials [Kir, Mac], which often run via double affine Hecke algebras[Che2]. This has lead to a whole family of Hecke algebras, with adjectives like de-generate, cyclotomic, rational, graded and (double) affine. We will only discuss onefurther member of this family, in paragraph 1.5.As already explained, Tits’ deformation theorem does not apply to affine Heckealgebras. But there is a substitute for the semisimplicity part of Theorem 1.1.Recall that a finite dimensional algebra A is semisimple (i.e. a direct sum of simplealgebras) if and only its Jacobson radical Jac( A ) (the intersection of the kernels of allsimple modules) is zero. In that sense Theorem 1.1 admits a partial generalization,see [Sol1, Lemma 3.4] and [Mat2, (3.4.5)]: Lemma 1.7.
The Jacobson radical of an affine Hecke algebra H ( R , λ, λ ∗ , q ) is zero. Presentations of affine Hecke algebras.
We will make the relations between the algebras in paragraphs 1.2 and 1.3 explicit.We start with the Bernstein presentation of an Iwahori–Hecke algebra of affine type.Let W aff be an affine Weyl group with Coxeter generators S aff , and write it as W aff = Z R (cid:111) W ( R ). Put R = (cid:0) Z R, R,
Hom Z ( Z R, Z ) , R ∨ , ∆ (cid:1) , where ∆ = { α ∈ R : s α ∈ S aff } . Consider an Iwahori–Hecke algebra H ( W aff , q ) withparameters q s ∈ C . Theorem 1.8. (Bernstein, see [Lus3, § )Suppose that q s (cid:54) = 0 for all s ∈ S aff . Pick λ ( α ) , λ ∗ ( α ) such that q s α = q λ ( α ) forall α ∈ R and q s (cid:48) α = q λ ∗ ( α ) when α ∨ ∈ R ∨ max . Then there exists a unique algebraisomorphism H ( W aff , q ) → H ( R , λ, λ ∗ , q ) such that: • it is the identity on H ( W, q ) , • for x ∈ Z R with (cid:104) x, α ∨ (cid:105) ≥ for all α ∈ ∆ , it sends q ( x ) − / T x to θ x .Here q ( x ) / is defined via (1.8) and q / s α = q λ ( α ) / , q / s (cid:48) α = q λ ∗ ( α ) / . Not every affine Hecke algebra is isomorphic to an Iwahori–Hecke algebra, forinstance C [ X ] is not. To be precise, an isomorphism as in Theorem 1.8 exists if andonly if the root datum R is that of an adjoint semisimple group. To compensatefor this difference in scope, we take another look at the structure of W ( R ) for anarbitrary based root datum R .We know from [Hum, § (cid:96) of ( W aff , S aff ) satisfies(1.17) (cid:96) ( w ) = number of hyperplanes H α,n that separate w ( A ) from A . We extend (cid:96) to a function W ( R ) → Z ≥ , by decreeing that (1.17) is valid for all w ∈ W ( R ). ThenΩ := { w ∈ W ( R ) : (cid:96) ( w ) = 0 } = stabilizer of A in W ( R )is a subgroup of W ( R ). The group Ω acts by conjugation on W aff and that actionstabilizes S aff (the set of reflections with respect to the walls of A ). Moreover, since FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 11 W aff acts simply transitively on the set of alcoves in X ⊗ Z R : W ( R ) = W aff (cid:111) Ω . Example 1.9.
Let R be of type GL n , as in (1.14). Then Ω is isomorphic to Z ,generated by ω = e (1 2 · · · n ). The action of ω on S aff = { s , s , . . . , s n − } is ωs i ω − = s i +1 (where s n means s ).Let q : S aff → C × be the parameter function determined by q , λ, λ ∗ . The con-ditions on λ and λ ∗ (before Definition 1.6) ensure that q is Ω-invariant. Hence theformula ω ( T w ) = T ωwω − defines an algebra automorphism of H ( W aff , q ). This gives a group action of Ω on H ( W aff , q ). Recall that the crossed product algebra H ( W aff , q ) (cid:111) Ω is the vectorspace H ( W aff , q ) ⊗ C C [Ω] with multiplication defined by ω · T w · ω − = ω ( T w ) . For w ∈ W aff , ω ∈ Ω we write T wω = T w ω ∈ H ( W aff , q ) (cid:111) Ω. The multiplicationrelations in H ( W aff , q ) (cid:111) Ω become(1.18) T v T w = T vw if (cid:96) ( vw ) = (cid:96) ( v ) + (cid:96) ( w ) , ( T s + 1)( T s − q s ) = 0 for s ∈ S aff . Now we can formulate the counterpart of Theorem 1.8.
Theorem 1.10. (Bernstein, see [Lus3, § )There is a unique algebra isomorphism H ( R , λ, λ ∗ , q ) → H ( W aff , q ) (cid:111) Ω such that • it is the identity on H ( W, q ) , • for x ∈ Z R with (cid:104) x, α ∨ (cid:105) ≥ for all α ∈ ∆ , it sends θ x to q ( x ) − / T x . The algebra H ( W aff , q ) (cid:111) Ω, with the multiplication rules (1.18), is called theIwahori–Matsumoto presentation of H ( R , λ, λ ∗ , q ).We conclude this paragraph with yet another, more geometric, presentation ofaffine Hecke algebras. Let T be the complex algebraic torus Hom Z ( X, C × ). Byduality Hom( T, C × ) = X , and the ring of regular functions O ( T ) is the groupalgebra C [ X ]. The group W acts naturally on X , and that induces actions on C [ X ]and on T . Definition 1.11.
The algebra H ( T, λ, λ ∗ , q ) is the vector space O ( T ) ⊗ C H ( W, q )with the multiplication rules • O ( T ) and H ( W, q ) are embedded as subalgebras, • for α ∈ ∆ and f ∈ O ( T ): f T s α − T s α ( s α · f ) = (cid:16) ( q λ ( α ) −
1) + θ − α (cid:0) q ( λ ( α )+ λ ∗ ( α )) / − q ( λ ( α ) − λ ∗ ( α )) / (cid:1)(cid:17) f − s α ( f ) θ − θ − α . Clearly the identification O ( T ) ∼ = C [ X ] induces an algebra isomorphism H ( T, λ, λ ∗ , q ) ∼ = H ( R , λ, λ ∗ , q ) . From the above presentation it is easy to find the centre of these algebras [Lus3]:(1.19) Z (cid:0) H ( R , λ, λ ∗ , q ) (cid:1) ∼ = Z (cid:0) H ( T, λ, λ ∗ , q ) (cid:1) = O ( T ) W = O ( T /W ) ∼ = C [ X ] W . An advantage of Definition 1.11 is that this presentation can also be used if T isjust known as an algebraic variety, without a group structure. In that situation H ( T, λ, λ ∗ , q ) can be studied without fixing a basepoint of T , it suffices to have the W -action and the elements θ − α ∈ O ( T ) × for α ∈ ∆. This is particularly handyfor affine Hecke algebras arising from Bernstein components and types for reductive p -adic groups.Even more flexibly, the above presentation applies when O ( T ) is replaced by areasonable algebra of differentiable functions on T , like rational functions, analyticfunctions or smooth functions.We already observed that Tits’ deformation theorem fails for affine Hecke algebras.Nevertheless, apart from Lemma 1.7 there is another analogue of Theorem 1.1,obtained by replacing the centre of an affine Hecke algebra by its quotient field.Let C ( X ) be the quotient field of C [ X ], that is, the field of rational functions onthe complex algebraic variety T . The action of W on T gives rise to the crossedproduct algebra C ( X ) (cid:111) W . The quotient field of Z (cid:0) H ( R , λ, λ ∗ , q ) (cid:1) ∼ = C [ X ] W is C ( X ) W , which is also the centre of C ( X ) (cid:111) W . We construct the algebra(1.20) C ( X ) W ⊗ C [ X ] W H ( T, λ, λ ∗ , q ) ∼ = C ( X ) ⊗ C [ X ] H ( T, λ, λ ∗ , q ) ∼ = C ( X ) ⊗ C H ( W, q ) . Here the multiplication comes from the description on the left, and it is the samealgebra as obtained from Definition 1.11 by substituting C ( X ) for O ( T ).For α ∈ ∆ we define an element ı ◦ s α of (1.20) by ı ◦ s α + 1 = q − λ ( α ) ( θ α − θ α + 1)( θ α − q ( λ ( α )+ λ ∗ ( α )) / )( θ α + q ( λ ( α ) − λ ∗ ( α )) / ) (1 + T s α ) . Proposition 1.12. [Lus3, § a ) The map s α (cid:55)→ ı ◦ s α extends to a group homomorphism W → (cid:0) C ( X ) W ⊗ C [ X ] W H ( T, λ, λ ∗ , q ) (cid:1) × : w (cid:55)→ ı ◦ w . ( b ) The map C ( X ) (cid:111) W → C ( X ) W ⊗ C [ X ] W H ( T, λ, λ ∗ , q ) : f ⊗ w (cid:55)→ f ı ◦ w is an algebra isomorphism. In particular ı ◦ w f ı ◦ w − = w ( f ) f ∈ C ( X ) , w ∈ W. Graded Hecke algebras.
Graded (affine) Hecke algebras were discovered in [Dri, Lus3]. They are simplifiedversions of affine Hecke algebras, more or less in the same way that a Lie algebra isa simplification of a Lie group.Let a be a finite dimensional Euclidean space and let W be a finite Coxetergroup action isometrically on a (and hence also on a ∗ ). Let R ⊂ a ∗ be a reducedroot system, stable under the action of W , such that the reflections s α with α ∈ R generate W . These conditions imply that W acts trivially on the orthogonalcomplement of R R in a ∗ .In contrast with the previous paragraphs, R does not have to be integral and W does not have to be crystallographic. Thus we are dealing with root systemsas in [Hum, § W -stable, no further condition. In particularthe upcoming construction applies equally well to Coxeter groups of type H , H or I ( m )2 . FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 13
Write t = a ⊗ R C and let S ( t ∗ ) = O ( t ) be the algebra of polynomial functions on t . We choose a W -invariant parameter function k : R → C and we let r be a formalvariable. We also fix a base ∆ of R . Definition 1.13.
The graded Hecke algebra H ( t , W, k, r ) is the vector space C [ W ] ⊗ C S ( t ∗ ) ⊗ C C [ r ] with the multiplication rules • C [ W ] and S ( t ∗ ) ⊗ C C [ r ] ∼ = O ( t ⊕ C ) are embedded as subalgebras, • C [ r ] is central, • the cross relation for α ∈ ∆ and ξ ∈ S ( t ∗ ): ξ · s α − s α · s α ( ξ ) = k ( α ) r ξ − s α ( ξ ) α . The grading is given by deg( w ) = 0 for w ∈ W and deg( x ) = deg( r ) = 2 for x ∈ t ∗ \ { } .Notice that for k = 0 Definition 1.13 yields the crossed product algebra H ( t , W, , r ) = C [ r ] ⊗ C S ( t ∗ ) (cid:111) W. Let R ∨ ⊂ a be the coroot system of R , so with (cid:104) α, α ∨ (cid:105) = 2 for all α ∈ R . For x ∈ t ∗ the following relation holds in H ( t , W, k, r ):(1.21) s α · x − s α ( x ) · s α = k ( α ) r (cid:104) x, α ∨ (cid:105) . In fact (1.21) can be substituted for the cross relation in Definition 1.13, that sufficesto determine the algebra structure uniquely.With such a simple presentation, it is no surprise that the graded Hecke algebras H ( t , W, k ) have more diverse applications than affine Hecke algebras. They appearin the representation theory of reductive groups over local fields, both in the p -case[BaMo1, BaMo2, Lus5] and in the real case [CiTr1, CiTr2]. Further, these gradedHecke algebras can be realized with Dunkl operators [Che1, Opd1], which enablesthem to act on many interesting function spaces.The above construction can be modified a little, and still produce the same alge-bra. Namely, fix α ∈ R and (cid:15) ∈ R > . For all w ∈ W we replace k ( wα ) by (cid:15)k ( wα )and wα by (cid:15)wα –that again gives a root system in the sense of [Hum, § R to be non-reduced, as long as we impose in additionthat k ( (cid:15)α ) = (cid:15)k ( α ) whenever (cid:15) > α, (cid:15)α ∈ R – that still gives the same gradedHecke algebras.Similarly, we can scale all parameters k ( α ) simultaneously. Namely, scalar multi-plication with z ∈ C × defines a bijection m z : t ∗ → t ∗ , which clearly extends to analgebra automorphism of S ( t ∗ ). From Definition 1.13 we see that it extends evenfurther, to an algebra isomorphism(1.22) m z : H ( t , W, zk, r ) → H ( t , W, k, r )which is the identity on C [ W ] ⊗ C C [ r ]. Notice that for z = 0 the map m z is well-defined, but no longer bijective. It is the canonical surjection H ( t , W, , r ) → C [ W ] ⊗ C C [ r ] . Algebras like H ( t , W, k, r ) are degenerations of affine Hecke algebras (the versionwhere q is a formal variable) and arise in the study of cuspidal local systems onunipotent orbits in complex reductive Lie algebras [Lus2, Lus4, Lus6, AMS2]. More often one encounters versions of H ( t , W, k, r ) with r specialized to a nonzerocomplex number. In view of (1.22) it hardly matters which specialization, so itsuffices to look at r (cid:55)→
1. The resulting algebra H ( t , W, k ) has underlying vectorspace C [ W ] ⊗ C S ( t ∗ ) and cross relations(1.23) ξ · s α − s α · s α ( ξ ) = k ( α )( ξ − s α ( ξ )) /α α ∈ ∆ , ξ ∈ S ( t ∗ ) . Like for affine Hecke algebras, we see from (1.23) that the centre of H ( t , W, k ) is(1.24) Z ( H ( t , W, k )) = S ( t ∗ ) W = O ( t /W ) . As a vector space, H ( t , W, k ) is still graded by deg( w ) = 0 for w ∈ W and deg( x ) = 2for x ∈ t ∗ \ { } . However, it is not a graded algebra any more, because (1.23) is nothomogeneous in the case ξ = α . Instead, the above grading merely makes H ( t , W, k )into a filtered algebra.The graded algebra obtained from this filtration is obtained by setting the righthand side of (1.23) equal to 0. In other words, the associated graded of H ( t , W, k )is the crossed product algebra H ( t , W,
0) = S ( t ∗ ) (cid:111) W. The algebras H ( t , W,
0) and H ( t , W, k ) with k (cid:54) = 0 are usually not isomorphic. Butthere is an analogue of Tits’ deformation theorem, similar to Proposition 1.12. Let Q ( S ( t ∗ )) be the quotient field of S ( t ∗ ), that is, the field of rational functions on t .It admits a natural W -action, and the centre of the crossed product Q ( S ( t ∗ )) (cid:111) W is Q (( S t ∗ )) W . Using (1.24) we construct the algebra Q ( S ( t ∗ )) W ⊗ S ( t ∗ ) W H ( t , W, k ),which as vector space equals Q ( S ( t ∗ )) ⊗ S ( t ∗ ) H ( t , W, k ) = Q ( S ( t ∗ )) ⊗ C C [ W ] . In there we have elements˜ ı s α = αα + k ( α ) (1 + s α ) − αα + k ( α ) s α − k ( α ) α + k ( α ) α ∈ ∆ . Proposition 1.14. [Lus3, § a ) The map s α (cid:55)→ ˜ ı s α extends to a group homomorphism W → (cid:0) Q ( S ( t ∗ )) W ⊗ S ( t ∗ ) W H ( t , W, k ) (cid:1) × : w (cid:55)→ ˜ ı w . ( b ) The map Q ( S ( t ∗ )) (cid:111) W → Q ( S ( t ∗ )) W ⊗ S ( t ∗ ) W H ( t , W, k ) : f ⊗ w (cid:55)→ f ˜ ı w is an algebra isomorphism. In particular ˜ ı w f ˜ ı w − = w ( f ) f ∈ Q ( S ( t ∗ )) , w ∈ W. Graded Hecke algebras can be decomposed like root systems and reductive Liealgebras. Let R , . . . , R d be the irreducible components of R . Write a ∗ i = span( R i ) ⊂ a ∗ , t i = Hom R ( a ∗ i , C ) and z = R ⊥ ⊂ t . Then t = t ⊕ · · · ⊕ t d ⊕ z . The inclusions W ( R i ) → W ( R ) , t ∗ i → t ∗ and z ∗ → t ∗ induce an algebra isomorphism(1.25) H ( t , W ( R ) , k ) ⊗ C · · · ⊗ C H ( t d , W ( R d ) , k ) ⊗ C O ( z ) −→ H ( t , W, k ) . Hence the representation theory of H ( t , W, k ) is more or less the product of therepresentation theories of the tensor factors in (1.25). The commutative algebra FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 15 O ( z ) ∼ = S ( z ∗ ) is of course very simple, so the study of graded Hecke algebra can bereduced to the case where the root system R is irreducible.2. Irreducible representations in special cases
The most elementary instance of an affine Hecke algebra is with empty root system R . The algebra associated to the root datum ( X, ∅ , Y, ∅ ) is just the group algebra C [ X ] of the lattice X . Its space of irreducible complex representations is(2.1) Irr( C [ X ]) = Irr( X ) = Hom Z ( X, C × ) = T. Since C [ X ] is a subalgebra of H (( X, R, Y, R ∨ ) , λ, λ ∗ , q ) for any additional data R, λ, λ ∗ and q , (2.1) is a good starting point for the representation theory of anyaffine Hecke algebra.We will discuss the affine Hecke algebras that appear most often, and we constructand classify all their irreducible representations.2.1. Affine Hecke algebras with q = 1 . Let R = ( X, R, Y, R ∨ , ∆) be an arbitrary based root datum and take the param-eters λ = λ ∗ = 0. Then q s = 1 for all s ∈ S aff , and H ( R , , , q ) = H ( R ,
1) = C [ X ] (cid:111) W. We denote the onedimensional representation of C [ X ] associated to t ∈ T by C t .Then ind C [ X ] (cid:111) W C [ X ] ( C t ) is a | W | -dimensional representation of C [ X ] (cid:111) W . Its restrictionto C [ X ] is Res C [ X ] (cid:111) W C [ X ] ind C [ X ] (cid:111) W C [ X ] ( C t ) = (cid:77) w ∈ W w C t ∼ = (cid:77) w ∈ W C w ( t ) . More generally, consider any C [ X ] (cid:111) W -representation ( π, V ) that is generated bythe subspace V t := { v ∈ V : π ( θ x ) v = x ( t ) v ∀ x ∈ X } . Then V = (cid:80) w ∈ W π ( w ) V t and π ( w ) V t = V w ( t ) . As V t ∩ V t (cid:48) = { } for t (cid:54) = t (cid:48) , V = ind C [ X ] (cid:111) W C [ X ] (cid:111) W t ( V t ) , where W t = { w ∈ W : w ( t ) = t } . By Frobenius reciprocityEnd C [ X ] (cid:111) W ( V ) ∼ = Hom C [ X ] (cid:111) W t ( V t , V ) =Hom C [ X ] (cid:111) W t (cid:0) V t , (cid:88) w ∈ W/W t V w ( t ) (cid:1) = End C [ X ] (cid:111) W t ( V t ) . Corollary 2.1.
The functor ind C [ X ] (cid:111) W C [ X ] (cid:111) W t induces an equivalences between the follow-ing categories: • C [ X ] (cid:111) W t -representations on which C [ X ] acts via the character t , • C [ X ] (cid:111) W -representations V that are generated by V t . The first category in Corollary 2.1 is naturally equivalent with the category of W t -representations. We conclude that, for every irreducible W t -representation ( ρ, V ρ ),the C [ X ] (cid:111) W -representation π ( t, ρ ) := ind C [ X ] (cid:111) W C [ X ] (cid:111) W t ( C t ⊗ V ρ ) is irreducible. For comparison with later results we point out that π ( t, ρ ) is a directsummand of the induced representationind C [ X ] (cid:111) W C [ X ] ( C t ) = ind C [ X ] (cid:111) W C [ X ] (cid:111) W t ( C t ⊗ C [ W t ])and that Res C [ X ] (cid:111) W C [ X ] π ( t, ρ ) = (cid:77) w ∈ W/W t C dim( V ρ ) w ( t ) . The next result goes back to Frobenius and Clifford, see [RaRa, Appendix] for amodern account.
Theorem 2.2.
Every irreducible C [ X ] (cid:111) W -representation is of the form π ( t, ρ ) for a t ∈ T and a ρ ∈ Irr( W t ) . Two such representations π ( t, ρ ) and π ( t (cid:48) , ρ (cid:48) ) areequivalent if and only if there exists a w ∈ W with t (cid:48) = w ( t ) and ρ (cid:48) = w · ρ . Here w · ρ = ρ ◦ Ad( w ) − : wW t w − → Aut C ( V ρ ). Theorem 2.2 involves a groupaction of W on the set ˜ T = { ( t, ρ ) : t ∈ T, ρ ∈ Irr( W t ) } . We call
T //W := ˜
T /W the extended quotient of T by W . Thus Theorem 2.2 gives a canonical bijection(2.2) T //W ←→ Irr( C [ X ] (cid:111) W ) = Irr (cid:0) H ( R , (cid:1) . On Irr( C [ X ] (cid:111) W ) we have the Jacobson topology, whose closed sets are { π ∈ Irr( C [ X ] (cid:111) W ) : S ⊂ ker π } for S ⊂ C [ X ] (cid:111) W. Via (2.2) we transfer this to a topology on
T //W . Then the natural maps
T /W → T //W T //W → T /WW t (cid:55)→ [ t, triv] [ t, ρ ] (cid:55)→ W t are continuous. The composition of (2.2) with
T //W → T /W is just the restrictionof an irreducible C [ X ] (cid:111) W -representation to C [ X ] W ∼ = O ( T /W ), in other words, itis the central character map.2.2.
Graded Hecke algebras with k = 0 . As in Paragraph 1.5, we let W be a finite Coxeter group acting isometrically ona finite dimensional Euclidean space a . When k = 0, we do not need a root system R ⊂ a ∗ to construct the algebra H ( t , W,
0) = O ( a ⊗ R C ) (cid:111) W = S ( t ∗ ) (cid:111) W. Considerations with Clifford theory, exactly as in the previous paragraph, lead to:
Theorem 2.3.
Every irreducible representation of O ( t ) (cid:111) W is of the form π ( ν, ρ ) = ind O ( t ) (cid:111) W O ( t ) (cid:111) W ν ( C ν ⊗ V ρ ) for some ν ∈ t , ( ρ, V ρ ) ∈ Irr( W ρ ) . Two such representations π ( ν, ρ ) and π ( ν (cid:48) , ρ (cid:48) ) are equivalent if and only if thereexists a w ∈ W with ν (cid:48) = w ( ν ) and ρ (cid:48) = w · ρ . Like for affine Hecke algebras with q = 1, π ( ν, ρ ) is a direct summand ofind O ( t ) (cid:111) W O ( t ) ( C ν ) = ind O ( t ) (cid:111) W O ( t ) (cid:111) W ν ( C t ⊗ C [ W ]) . Also, there is a canonical bijection t //W ←→ Irr( O ( t ) (cid:111) W ) = Irr (cid:0) H ( t , W, (cid:1) . FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 17
Iwahori–Hecke algebras of type (cid:102) A . Consider the based root datum R = (cid:0) X = Z , R = {± } , Y = Z , R ∨ = {± } , ∆ = { α } = { } (cid:1) . It has an affine Weyl group W aff = W ( R ) of type (cid:102) A , with Coxeter generators S aff = { s α , s (cid:48) α : x (cid:55)→ − x } . Affine Hecke algebras of the H ( R , λ, λ ∗ , q ), for variousparameters λ ( α ) and λ ∗ ( α ), appear often in the representation theory of classical p -adic groups [GoRo, MiSt]. We will work out the irreducible representations of H ( R , λ, λ ∗ , q ) in detail. To avoid singular cases, we assume throughout this para-graph that(2.3) q λ ( α ) (cid:54) = − , q λ ∗ ( α ) (cid:54) = − , q λ ( α )+ λ ∗ ( α ) (cid:54) = 1 . Recall that the case λ ( α ) = λ ∗ ( α ) = 0 was already discussed in Paragraph 2.1. Weabbreviate q / = q λ ( α ) / , q / = q λ ∗ ( α ) / and H = H ( R , λ, λ ∗ , q ) . It is not difficult to see [Mat1] that every irreducible H -representation is a quotientof ind H C [ X ] ( C t ) for some t ∈ T . By (1.16) this induced representation has dimensiontwo. Using the Iwahori–Matsumoto presentation H ( W aff , q ), two onedimensionalrepresentations can be written down immediately. Firstly the trivial representation,given by triv( T s α ) = q λ ( α ) = q , triv( T s (cid:48) α ) = q λ ∗ ( α ) = q , and secondly the Steinberg representation, defined bySt( T s α ) = − , St( T s (cid:48) α ) = − . When H is the Iwahori-spherical Hecke algebra of SL over a p -adic field F , thesetwo representations correspond to the trivial and the Steinberg representations of SL ( F ) – as their names already suggested.Via evaluation at 1 ∈ Z , we identify T = Hom Z ( X, C × ) with C × . By Theorem1.8 θ = q − / q − / T s (cid:48) α T s α . For the trivial and Steinberg representations that meanstriv( θ ) = q − / q − / triv( T s (cid:48) α T s α ) = q / q / , St( θ ) = q − / q − / St( T s (cid:48) α T s α ) = q − / q − / . Therefore, as C [ X ]-representations:(2.4) triv | C [ X ] = C q / q / and St | C [ X ] = C q − / q − / . Theorem 2.4. ( a ) The H -representation ind H C [ X ] ( C t ) is irreducible for all t ∈ C × \ (cid:8) q / q / , q − / q − / , − q / q − / , − q − / q / (cid:9) . ( b ) For t as in part (a), ind H C [ X ] ( C t ) is isomorphic with ind H C [ X ] ( C t − ) . There are nofurther relations between the irreducible representations ind H C [ X ] ( C t ) . ( c ) The algebra H has precisely four other irreducible representations: triv , St andtwo that we call π ( − , triv) , π ( − , St) . They have dimension one and fit in short exact sequences → St → ind H C [ X ] ( C q / q / ) → triv → , → triv → ind H C [ X ] ( C q − / q − / ) → St → , → π ( − , St) → ind H C [ X ] ( C − q − / q / ) → π ( − , triv) → , → π ( − , triv) → ind H C [ X ] ( C − q / q − / ) → π ( − , St) → . Remark.
By the conditions (2.3), the four special values of t are all different,except that the last two coincide if q = q . Proof. (a) By (1.16) ind H C [ X ] ( C t ) = H ( W, q ) as H ( W, q )-module. Since q (cid:54) = −
1, thealgebra H ( W, q ) is semisimple, and isomorphic with C [ W ] ∼ = C ⊕ C . The quadraticrelation (1.1) points us to the minimal central idempotents in H ( W, q ): p + := ( T s α + T e )(1 + q ) − and p − := ( T s α − q T e )(1 + q ) − . Then C p + and C p − are the only nontrivial H ( W, q )-submodules of ind H C [ X ] ( C t ). Itfollows that, whenever ind H C [ X ] ( C t ) is irreducible as H -module, C p + or C p − is an H -submodule. We test for which t ∈ T this happens. The cross relation in H gives θ ( T s α + T e ) = θ + T s α θ − + (cid:16) ( q λ ( α ) −
1) + θ − α (cid:0) q ( λ ( α )+ λ ∗ ( α )) / − q ( λ ( α ) − λ ∗ ( α )) / (cid:1)(cid:17) θ − θ − θ − θ − = T s α θ − + θ (cid:0) ( q −
1) + θ − ( q / q / − q / q − / ) (cid:1) θ = T s α θ − + q θ + q / ( q / − q − / ) . In ind H C [ X ] ( C t ) we get(2.5) θ (1 + q ) p + = θ ( T s α + T e ) = T s α · t − + q t + q / ( q / − q − / )= t − ( T s α + T e ) + (cid:0) tq − t − + q / ( q / − q − / ) (cid:1) T e = t − ( T s α + T e ) + q / ( tq / − t − q − / + q / − q − / ) T e . This can only be a scalar multiple of p + if tq / − t − q − / + q / − q − / = 0, andthat happens only if q / = − tq / or q / = t − q − / .Similarly we compute in H : θ ( T s α − q T e ) = T s α θ − − θ + q / ( q / − q − / ) . In ind H C [ X ] ( C t ) that leads to(2.6) θ (1 + q ) p − = θ ( T s α − q T e ) = T s α · t − − t + q / ( q / − q − / )= t − ( T s α − q T e ) + (cid:0) q t − − t + q / ( q / − q − / ) (cid:1) T e = t − ( T s α − q T e ) + q / ( t − q / − tq − / + q / − q − / ) T e . If this is a scalar multiple of p − , then t − q / − tq − / + q / − q − / = 0, whichmeans that q / = − t − q / or q / = tq − / .We conclude that for t ∈ C × \ (cid:8) q / q / , q − / q − / , − q / q − / , − q − / q / (cid:9) FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 19 neither C p + nor C p − is an H -submodule of ind H C [ X ] ( C t ), so that ind H C [ X ] ( C t ) is irre-ducible. On the other hand, when t equals one of these special values, the above cal-culations in combination with the fact that θ generates C [ X ] imply that ind H C [ X ] ( C t )does have a onedimensional H -submodule.(b) Consider the element f α := θ ( q −
1) + q / ( q / − q − / ) θ − θ − of the quotient field C ( X ) of C [ X ] = O ( T ). It lies in the version of H = H ( T, λ, λ ∗ , q )obtained from Definition 1.13 by replacing O ( T ) with C ( X ). By direct calculationin that algebra: θ x ( T s α − f α ) = ( T s α − f α ) θ x for all x ∈ X = Z . Although f α / ∈ C [ X ], it has a well-defined action on ind H C [ X ] ( C t ), provided that( θ − θ − )( t ) (cid:54) = 0, or equivalently t / ∈ { , − } . For v ∈ C t \ { } , the element( T s α − f α ) v ∈ ind H C [ X ] ( C t ) \ { } satisfies θ x ( T s α − f α ) v = ( T s α − f α ) θ − x v = ( T s α − f α ) θ − x ( t ) v = t − ( x )( T s α − f α ) v. Hence as C [ X ]-modules(2.7) ind H C [ X ] ( C t ) = C t ⊕ C t − for all t ∈ C × \ { , − } . By Frobenius reciprocity, for such t :Hom H (cid:0) ind H C [ X ] ( C t − ) , ind H C [ X ] ( C t ) (cid:1) ∼ = Hom C [ X ] (cid:0) C t − , ind H C [ X ] ( C t ) (cid:1) = Hom C [ X ] (cid:0) C t − , C t ⊕ C t − (cid:1) ∼ = C . In particular this shows thatind H C [ X ] ( C t − ) ∼ = ind H C [ X ] ( C t )whenever these representations are irreducible (for t = ± H C [ X ] ( C q / q / ) has a subrepresentation C p − with θ p − = q − / q − / p − . Further, by (1.1)(2.8) ( T s α + T e ) p − = 0 , so T s α p − = − p − . As T s α and θ generate H , it follows that here C p − is the Steinberg representation.By (2.7) ind H C [ X ] ( C q / q / ) / C p − ∼ = C q / q / as C [ X ]-representation. Alsoind H C [ X ] ( C q / q / ) / C p − ∼ = H ( W, q ) / C p − ∼ = C p + as H ( W, q )-representation. From ( T s α − q T e ) p + = 0 we see that T s α p + = q p + .Again, since θ and T s α generate H , we can conclude that ind H C [ X ] ( C q / q / ) / C p − isthe trivial representation of H . The calculations around (2.6) show that ind H C [ X ] ( C − q − / q / ) contains a subrep-resentation π ( − , St) = C p − with(2.9) θ p − = − q / q − / p − . By (2.8) it has the same restriction to H ( W, q ) as the Steinberg representation, whichexplains our notation π ( − , St) for this C p − . The quotient(2.10) ind H C [ X ] ( C − q − / q / ) / C p − equals C − q − / q / as C [ X ]-representation.As H ( W, q )-representation it is isomorphic to C p + ∼ = triv, and therefore we write π ( − , triv) = ind H C [ X ] ( C − q / q − / ) / C p − . Analogous considerations apply to ind H C [ X ] ( C q − / q − / ) and ind H C [ X ] ( C − q / q − / ). (cid:3) An important special case is q = q = q . When F is a non-archimedean localfield with with residue field of order q , H ( W aff , q ) = H ( R , q ) arises as the Iwahori-spherical Hecke algebra of SL ( F ).The algebra H ( R , q ) is the simplest example of a true affine Hecke algebra, notisomorphic to some more elementary kind of algebra. For this algebra Theorem 2.4says: • The H ( R , q )-representation ind H ( R , q ) C [ X ] ( C t ) is irreducible for all t ∈ C × \ { q , q − , − } . • ind H ( R , q ) C [ X ] ( C t ) ∼ = ind H ( R , q ) C [ X ] ( C t − ) for all t ∈ C × \ { q , q − } . • ind H ( R , q ) C [ X ] ( C − ) = π ( − , triv) ⊕ π ( − , St), with π ( − , triv) and π ( − , St)irreducible and inequivalent. • There are only two other irreducible H ( R , q )-representations, triv and St,which both occur as subquotients of ind H ( R , q ) C [ X ] ( C q ) and of ind H ( R , q ) C [ X ] ( C q − ).This classification works for almost all q ∈ C × , only 1 and − H ( R , −
1) the trivial representationcoincides with π ( − , triv) and the Steinberg representation coincides with π ( − , St).Consequently H ( R , −
1) has only two onedimensional representations. Apart fromthat, the above statements are valid when q = − H ( R , q ) look considerably simpler thanthose for H ( R , λ, λ ∗ , q ), in the end the latter is hardly more difficult. In termsof the induced representations ind H C [ X ] ( C t ), the only differences occur when t ∈ (cid:8) − , − q / q − / , − q − / q / (cid:9) .2.4. Affine Hecke algebras of type GL n . The root datum of type GL n is R n = ( Z n , A n − , Z n , A n − , ∆ n )with A n − = { e i − e j : 1 ≤ i, j ≤ n, i (cid:54) = j } and ∆ n − = { e i − e i +1 : i = 1 , , . . . n − } .Via t (cid:55)→ ( t ( e ) , . . . , t ( e n )) we identify T with ( C × ) n . We saw in Section 1 that W aff = { x ∈ Z n : x + · · · + x n = 0 } (cid:111) S n ,S aff = { s i = s e i − e i +1 : i = 1 , . . . , n − } ∪ { s } , Ω = (cid:104) ω (cid:105) ∼ = Z , ω ( x ) = e + (1 2 · · · n ) x, FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 21 where s ( x ) = x + (1 − (cid:104) x, e − e n (cid:105) )( e − e n ). All the simple affine reflections from S aff are Ω-conjugate, so q s = q s (cid:48) for s, s (cid:48) ∈ S aff . Call this parameter q and considerthe affine Hecke algebra H n ( q ) := H ( R n , q )of type GL n . The primary importance of such algebras is that they describe everyBernstein block in the representation theory of GL n ( F ) for a non-archimedean localfield F [BuKu1]. In all those cases q is a power of a prime number, but just as affineHecke algebra that is not necessary. We do not even have to require that q ∈ R > ,the representation theory of H n ( q ) looks the same for every q ∈ C × which is not aroot of unity.The irreducible representations of GL n ( F ) were classified (in terms of supercusp-idal representations) by Zelevinsky [Zel, BeZe]. That classification follows a combi-natorial pattern involving certain ”segments, and the irreducible representations of H n ( q ) exhibit the same pattern. We formulate their classification in terms intrinsicto Hecke algebras. The algebra H ( q ) = C [ Z ] has already been discussed, so wemay assume that n ≥ S aff is nonempty).Like in Paragraph 2.3 we start with the trivial and the Steinberg representations.By definition(2.11) triv( T s ) = q , St( T s ) = − S aff . This does not yet determine the values of these representations on T ω , to fix thatone requires in additiontriv | C [ X ] = C t + , t + = (cid:0) q ( n − / , q ( n − / , . . . , q (1 − n ) / (cid:1) , St | C [ X ] = C t − , t − = (cid:0) q (1 − n ) / , q (3 − n ) / , . . . , q ( n − / (cid:1) . The terminology is motivated by Iwahori-spherical representation of GL n over a p -adic field F : triv and St correspond to the epynomous representations of GL n ( F ).In contrast with the Iwahori–Hecke algebra of type (cid:102) A , the trivial and Steinbergrepresentations of H n ( q ) come in a families of representations, parametrized by C × .This is implemented as follows. For z ∈ C × we put t z = ( z, z, . . . , z ) ∈ T S n . Wedefine the H n ( q )-representation triv ⊗ t z by (2.11) and(2.12) (triv ⊗ t z ) | C [ X ] = C t + t z . This is possible because α ( t z ) = 1 for all α ∈ R , so that t z ∈ Hom(
X, C × ) is trivialon X ∩ W aff . Similarly we define the onedimensional H n ( q )-representation St ⊗ t z ,by requiring (2.11) and(2.13) (St ⊗ t z ) | C [ X ] = C t − t z . For H ( q ) the above onedimensional representations, in combination with the in-duced representations ind H ( q ) C [ X ] ( C t ), already exhaust Irr( H ( q )). With argumentsvery similar to those in Paragraph 2.3 one can show: Theorem 2.5.
The H ( q ) -representation ind H ( q ) C [ X ] ( C t ) is irreducible for all t ∈ T that are not of the form t + t z = ( q / z, q − / z ) or t − t z = ( q − / z, q / z ) .This representation is isomorphic to ind H ( q ) C [ X ] ( C t − ) , but apart from that there areno relations between the irreducible representations of the form ind H ( q ) C [ X ] ( C t ) . The only other irreducible H ( q ) -representations are triv ⊗ t z and St ⊗ t z with z ∈ C × . They are mutually inequivalent and fit in short exact sequences → St ⊗ t z → ind H ( q ) C [ X ] ( C ( q / z, q − / z ) ) → triv ⊗ t z → , → triv ⊗ t z → ind H ( q ) C [ X ] ( C ( q − / z, q / z ) ) → St ⊗ t z → . The irreducible representations from Theorem 2.5 come in three kinds, but thereis a natural way to gather them in two families: • The twists St ⊗ t z of the Steinberg representation, parametrized by t z ∈ T S . • A family parametrized by
T /S , which for almost all W t ∈ T /W has themember ind H ( q ) C [ X ] ( C t ). When that representation happens to be reducible,we take the unique representative t of W t with | α ( t ) | ≥ W t the unique irreducible quotient of ind H ( q ) C [ X ] ( C t ).The irreducible representations of H n ( q ) with n ≥ • Choose a partition (cid:126)n = ( n , n , . . . , n d ) of n , where n i ≥ n i ) need not be monotone). • The algebra d (cid:79) i =1 H n i ( q ) = d (cid:79) i =1 H (cid:0) ( Z n i , A n i − , Z n i , A n i − , ∆ n i − ) , q (cid:1) is naturally a subalgebra of H n ( q ), with the same commutative subalgebra C [ X ] = C [ Z n ] = ⊗ ni =1 C [ Z n i ]. • For every i we pick z i ∈ Irr( C [ Z n i ]) S ni ∼ = (cid:0) ( C × ) n i (cid:1) S ni ∼ = C × , and we construct the irreducible H n i ( q )-representation St ⊗ z i . (It corre-sponds to a segment in Zelevinsky’s setup.) • Then (cid:2) di =1 (St ⊗ z i ) is an irreducible representation of (cid:78) di =1 H n i ( q ) and π ( (cid:126)n, (cid:126)z ) := ind H n ( q ) (cid:78) di =1 H ni ( q ) (cid:16) (cid:2) di =1 (St ⊗ z i ) (cid:1) is an H n ( q )-representation. • For almost all (cid:126)z = ( z i ) di =1 , π ( (cid:126)n, (cid:126)z ) is irreducible and depends only on theorbit of (cid:126)z under N S n (cid:0) (cid:81) di =1 S n i (cid:1) . Here (cid:81) di =1 S n i fixes ( z i ) di =1 and N S n (cid:16) d (cid:89) i =1 S n i (cid:17)(cid:46) d (cid:89) i =1 S n i ∼ = (cid:89) m ≥ S ( { i : n i = m } ) . • When π ( (cid:126)n, (cid:126)z ) is reducible, we pick a representative (cid:126)z for N S n ( (cid:81) di =1 S n i ) (cid:126)z such that | z i | ≥ | z j | whenever n i = n j and i ≤ j . • For such (cid:126)z it follows from the Langlands classification that π ( (cid:126)n, (cid:126)z ) has aunique irreducible quotient. That is the irreducible H n ( q )-representationassociated with ( (cid:126)n, (cid:126)z ). • The irreducible H n ( q )-representations assigned to ( (cid:126)n, (cid:126)z ) and ( (cid:126)n (cid:48) , (cid:126)z (cid:48) ) are equiv-alent if and only if ( (cid:126)n, (cid:126)z ) and ( (cid:126)n (cid:48) , (cid:126)z (cid:48) ) are S n -associate. FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 23
In the above parametrization z i ∈ (cid:0) ( C × ) n i (cid:1) S ni , so (cid:126)z can be regarded as a diagonalmatrix in GL n ( C ). The partition (cid:126)n determines a Levi subgroup M = GL n ( C ) × · · · × GL n d ( C ) of GL n ( C ) , and (cid:126)z ∈ Z ( M ). Let u m be a regular unipotent element of GL m ( C ) (it is unique upto conjugation). Then (cid:126)u := ( u n , u n , . . . , u n d )is a regular unipotent element of M , and (cid:126)z(cid:126)u is the Jordan decomposition of anelement of GL n ( C ). Up to conjugacy, every element of GL n ( C ) has this shape, forsome (cid:126)z (unique up to S n -association) and (cid:126)u . This setup leads to: Theorem 2.6.
There exists a canonical bijection between the following sets: • conjugacy classes in GL n ( C ) , • S n -association classes of data ( (cid:126)n, (cid:126)z ) , where (cid:126)n = ( n i ) is a partition of n and z i ∈ (cid:0) ( C × ) n i (cid:1) S ni , • Irr( H n ( q )) . Representation theory
Parabolic induction.
In the representation theory of reductive groups, a pivotal role is played by para-bolic induction. An analogous operation exists for affine Hecke algebras, and it willbe crucial in large parts of this paper.Given a based root datum R = ( X, R, Y, R ∨ , ∆) and a subset P ⊂ ∆, we canform the based root datum R P = ( X, R P , Y, R ∨ P , P ) . Here R P = Q P ∩ R is a standard parabolic root subsystem of R , with dual rootsystem R ∨ P = Q P ∨ ∩ R ∨ . We record the special cases R ∆ = R and R ∅ = ( X, ∅ , Y, ∅ , ∅ ) . Let W P be the Weyl group of R P . Any parameter functions λ, λ ∗ for R restrict toparameter functions for R P , and H P = H ( R P , λ, λ ∗ , q )is a subalgebra of H = H ( R , λ, λ ∗ , q ). This corresponds to the notion of a parabolicsubgroup P of a reductive group G , and simultaneously to the notion of a Levifactor of P – for affine Hecke algebras the unipotent radical of P is more or lessautomatically divided out.Notice that H and H P share the same commutative subalgebra C [ X ] = O ( T ).By (1.16), as vector spaces(3.1) H = H ( W, q ) ⊗ H ( W P ,q ) H ( W P , q ) ⊗ C C [ X ] = H ( W, q ) ⊗ H ( W P ,q ) H P . Parabolic induction for representations of affine Hecke algebras is the functorind HH P : Mod( H P ) → Mod( H ) , for any parabolic subalgebra H P of H . We also have a ”parabolic restriction” func-tor, that is just restriction of H -modules to H .The link with parabolic induction for reductive p -adic groups is made precise in[Sol7, § the Jacquet restriction functor (but with respect to a parabolic subgroup oppositeto the one used for induction).In practice we often precompose parabolic induction with inflations of represen-tations from a quotient algebra of H P . To define it, we write X P = X (cid:14)(cid:0) X ∩ ( P ∨ ) ⊥ (cid:1) Y P = Y ∩ P ⊥ X P = X/ ( X ∩ Q P ) Y P = Y ∩ Q P ∨ R P = ( X P , R P , Y P , R P , P ) . In terms of reductive groups, the semisimple root datum R P corresponds to the max-imal semisimple quotient of a parabolic subgroup P of G . The parameter functions λ and λ ∗ remain well-defined for R P , so there is an affine Hecke algebra H P = H ( R P , λ, λ ∗ , q ) . In particular we have H ∅ = H (0 , ∅ , , ∅ , ∅ , λ, λ ∗ , q ) = C and H ∆ = H ( X ∆ , R, Y ∆ , R ∨ , ∆ , λ, λ ∗ , q ) . From Definition 1.6 we see that the quotient map X → X P : x (cid:55)→ x P induces a surjective algebra isomorphism(3.2) H P → H P : θ x T w (cid:55)→ θ x P T w . Via this quotient map we will often (implicitly) inflate H P -representations to H P -representations. To incorporate representations of H P that are nontrivial on { θ x : x ∈ X ∩ ( P ∨ ) ⊥ } , we need more flexibility. Write T P = Hom Z ( X P , C × ) , T P = Hom Z ( X P , C × ) , so that T P T P = T and T P ∩ T P is finite. Every t ∈ T P gives rise to an algebraautomorphism ψ t : H P → H P θ x T w (cid:55)→ x ( t ) θ x T w . This operation corresponds to twisting representations of a reductive group by anunramified character. For any H P -representation ( π, V ) and any t ∈ T P , we canform the H P -representation ψ ∗ t inf P ( π ) = π ◦ ψ t : θ x T w (cid:55)→ π ( x ( t ) θ x P T w ) . Definition 3.1.
The parabolically induced representation associated to P ⊂ ∆,( σ, V σ ) ∈ Mod( H P ) and t ∈ T P is π ( P, σ, t ) = ind HH P ( σ ◦ ψ t ) . By (3.1) the vector space underlying π ( P, σ, t ) is H ( W, q ) ⊗ H ( W P ,q ) V σ , and it has dimension [ W : W P ] dim( V σ ). Let us discuss how parabolic inductionworks out for the subalgebra C [ X ] of H . Definition 3.2.
Let ( π, V ) be an H -representation. For t ∈ T we write V t = (cid:8) v ∈ V | ∃ N ∈ N : ( π ( θ x ) − x ( t )) N v = 0 (cid:9) . When V t (cid:54) = 0, we call t a C [ X ]-weight of π , and V t its generalized weight space. Wedenote the set of C [ X ]-weights of ( π, V ) by Wt( π ) or Wt( V ). FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 25
For all t ∈ Wt( π ) the space V t contains a vector v (cid:48) (cid:54) = 0 with π ( θ x ) v (cid:48) = x ( t ) v (cid:48) ∀ x ∈ X, which explains why we call V t a weight space and not a generalized weight space.If V has finite dimension, then we can triangularize the commuting operators π ( θ x )simultaneously, and we find(3.3) V = (cid:77) t ∈ T V t . An infinite dimensional H -representation does not necessarily have any C [ X ]-weight.Recall that the centre of H is Z ( H ) = C [ X ] W = O ( T /W ). Hence, whenever π admits a central character cc( π ), we havecc( π ) = W Wt( π ) ∈ T /W.
The set of C [ X ]-weights of a representation behaves well under parabolic induction.To describe the effect, let W P be the set of shortest length representatives for W/W P . Lemma 3.3. ( a ) Let π be a finite dimensional H P -representation. Then the C [ X ] -weights of ind HH P ( π ) are the elements w ( t ) with t ∈ Wt( π ) and w ∈ W P . ( b ) Let σ be a finite dimensional H P -representation and let s ∈ T P . Then Wt( π ( P, σ, s )) = { w ( st ) : t ∈ Wt( σ ) , w ∈ W P } . Proof. (a) is a consequence of the proof of [Opd2, Proposition 4.20].(b) Cleary Wt( σ ◦ ψ s ) = s Wt( σ ). Combine with part (a). (cid:3) Since H has finite rank as a module over its centre C [ X ] W , every irreducible H -representation has finite dimension. Hence π admits at least one C [ X ]-weight, say t . By Frobenius reciprocityHom H (ind H C [ X ] ( C t ) , π ) ∼ = Hom C [ X ] ( C t , π ) (cid:54) = 0 . We conclude that:
Corollary 3.4.
Every irreducible H -representation is a quotient of ind H C [ X ] ( C t ) forsome t ∈ T . Most of the time ind H C [ X ] ( C t ) is itself irreducible. To make that precise, considerthe following rational functions on T :(3.4) c α = (cid:0) θ α − q ( − λ ∗ ( α ) − λ ( α )) / (cid:1)(cid:0) θ α + q ( λ ∗ ( α ) − λ ( α )) / (cid:1) ( θ α − θ α + 1) α ∈ R. There are a few ways in which c α can simplify: • if λ ( α ) = λ ∗ ( α ) = 0, then c α = 1, • if λ ( α ) = λ ∗ ( α ) (cid:54) = 0, then c α = ( θ α − q − λ ( α ) )( θ α − − . Theorem 3.5. [Kat1, Theorem 2.2]
Let t ∈ T . The H -representation ind H C [ X ] ( C t ) is irreducible if and only if • c α ( t ) (cid:54) = 0 for all α ∈ R and • W t is generated by { s α : α ∈ R, s α ( t ) = t, c − α ( t ) = 0 } . The parabolically induced representations(3.5) ind H C [ X ] ( C t ) = π ( ∅ , triv , t ) t ∈ T can all be realized on the same vector space H ( W, q ). In fact, they are isomorphic to H ( W, q ) as H ( W, q )-modules. In principle, the entire representation theory of H canbe uncovered by analysing the family of representations (3.5). We already did thatsuccessfully for W aff of type (cid:102) A in paragraph 2.3. However, this direct approach isvery difficult in general. Indeed, while the irreducible representations of H have beenclassified in several ways, the finer structure of ind H C [ X ] ( C t ) (e.g. a Jordan–H¨oldersequence or the multiplicity with which irreducible representations appear) is notalways known.3.2. Tempered representations.
An admissible representation of a reductive group G over a local field is temperedif all its matrix coefficents have moderate growth on G , see [Wal, § III.2] and [Kna, § VII.11]. This notion has several uses: • the irreducible tempered G -representations form precisely the support of thePlancherel measure of G , • the Langlands classification of irreducible admissible G -representations interms of irreducible tempered representations of Levi subgroups of G , • for general harmonic analysis on G , e.g. the Plancherel isomorphism.Analogous of all these well-known results have been established for affine Heckealgebras, see [Opd2, DeOp1]. In this paragraph we will discuss the second of theabove three items.Recall that every (finite dimensional) H -module ( π, V ) has a set of C [ X ]-weightsWt( π ) ⊂ T = Hom Z ( X, C × ). To formulate the condition for temperedness in termsof weights, it will be convenient to abbreviate a = Y ⊗ Z R , t = Y ⊗ Z R = Lie( T ) , a ∗ = X ⊗ Z R , t ∗ = X ⊗ Z R . The complex torus T admits a polar decomposition(3.6) T = Hom Z ( X, S ) × Hom Z ( X, R > ) = T un × exp( a ) . Here the unitary part T un is the maximal compact subgroup of T and the positivepart exp( a ) is the identity component of the maximal real split subtorus of T . Noticethat Lie( T un ) = i a = Y ⊗ Z i R ⊂ t = Lie( T ) . For any t ∈ T we write | t | for the homomorphism x (cid:55)→ | t ( x ) | . Then t = t | t | − | t | isthe polar decomposition of t .The acute positive cones in a are a + = { ν ∈ a : (cid:104) α, ν (cid:105) ≥ ∀ α ∈ ∆ } , a ++ = { ν ∈ a : (cid:104) α, ν (cid:105) > ∀ α ∈ ∆ } . We define a ∗ , + and a ∗ , ++ similarly, so in particular X + = X ∩ a ∗ , + . Next we wehave the obtuse negative cones in a :(3.7) a − = { ν ∈ a : (cid:104) δ, ν (cid:105) ≤ ∀ δ ∈ a ∗ , + } = { (cid:80) α ∈ ∆ x α α ∨ : x α ≤ } , a −− = { ν ∈ a : (cid:104) δ, ν (cid:105) < ∀ δ ∈ a ∗ , + \ { }} . Via the exponential map exp : t → T we get T + = exp( a + ) , T ++ = exp( a ++ ) , T − = exp( a − ) , T −− = exp( a −− ) . FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 27
Definition 3.6.
A finite dimensional H -representation ( π, V ) is tempered if thefollowing equivalent conditions are satisfied: • | t ( x ) | ≤ t ∈ Wt( π ) , x ∈ X + , • Wt( V ) ⊂ T un T − , • | Wt( V ) | ⊂ T − . Example 3.7. • Consider the root datum R n of type GL n . Then a = a ∗ = R n , a + = a ∗ , + = { ν ∈ R n : ν ≥ ν ≥ · · · ≥ ν n } , a − = { ν ∈ R n : ν ≤ , ν + ν ≤ , . . . , ν + ν + · · · + ν n − ≤ , ν + ν + · · · + ν n = 0 } . The Steinberg representation has (cid:0) q (1 − n ) / , q (3 − n ) / , . . . , q ( n − / (cid:1) as its only C [ X ]-weight, so it is tempered when q ≥
1. On the other hand, the trivial H n ( q )-representation satisfies Wt(triv) = (cid:8) ( q ( n − / , q ( n − / , . . . , q (1 − n ) / ) (cid:9) ,so it is tempered when q ≤ • For q = 1, the O ( T )-weights of any irreducible representation of H ( R ,
1) = C [ X (cid:111) W ] form a full W -orbit in T , see Paragraph 2.1. The W -orbit oflog | t | ∈ a can only be contained in a − if log | t | = 0, that is, | t | = 1. Hence theirreducible tempered representations of C [ X (cid:111) W ] are precisely the irreducibleconstituents of the modules ind C [ X (cid:111) W ] C [ X ] C t with t ∈ T un .With Lemma 3.3 one can show (see [Sol4, Lemma 3.1.1] and [AMS3, Lemma 2.4.c]): Proposition 3.8.
Let ( π, V ) be a finite dimensional representation of H P , for some P ⊂ ∆ . Then π is tempered ⇐⇒ ind HH P ( π ) is tempered. For P = ∅ we have R = ∅ and T + = T − = { } . Then Proposition 3.8 says thatind H C [ X ] ( C t ) is tempered if and only if t ∈ T un .In the Langlands classification we need a somewhat more general kind of H -representation, for which we merely require that it becomes tempered upon restric-tion to H ( W aff , q ). This can also be formulated with a more relaxed condition onthe weights. This involves the Lie subgroup T ∆ of T , whose Lie algebra is identifiedwith t ∆ := R ⊥ ⊂ t . Definition 3.9.
A finite dimensional H -representation ( π, V ) is essentially temperedif the following equivalent conditions are satisfied: • | t ( x ) | ≤ t ∈ Wt( π ) , x ∈ X + ∩ W aff , • Wt( V ) ⊂ T ∆ T un T − , • | Wt( V ) | ⊂ T ∆ T − .When the root datum R is semisimple, essentially tempered is the same as tempered. Lemma 3.10. (see [Sol1, Lemma 3.5] and [Sol7, Lemma 2.3])
For any irreducible essentially tempered H -representation π , there exists t ∈ T ∆ such that π ◦ ψ t is tempered and arises by inflation from a representation of H ∆ . Example 3.11.
Consider R n and H n ( q ) with q >
1. Then X + ∩ W aff = { x ∈ Z n : x ≥ x ≥ · · · ≥ x n , x + x + · · · + x n = 0 } . For z ∈ C × the H n ( q )-representation St ⊗ t z from (2.13) has a unique C [ X ]-weight (cid:0) z q (1 − n ) / , z q (3 − n ) / , . . . , z q ( n − / (cid:1) ∈ T ∆ T un T − . It is essentially tempered for all z ∈ C , and tempered if and only if | z | = 1.In the Langlands classification we employ irreducible representations of H P , where P ⊂ ∆. We need some further notations: t P = Y P ⊗ Z C = Lie( T P ) , a P = Y P ⊗ Z i R = Lie( T P un ) , a P, + = { ν ∈ a P : (cid:104) α, ν (cid:105) ≥ ∀ α ∈ ∆ \ P } ,T P, + = exp( a P, + ) = { t ∈ T P ∩ exp( a ) : | t ( α ) | ≥ ∀ α ∈ ∆ \ P } , a P, ++ = { ν ∈ a P : (cid:104) α, ν (cid:105) > ∀ ∆ \ P } T P, ++ = exp( a P, ++ ) . We say that ( π, V ) ∈ Irr( H P ) is in positive position if cc( π ) = tW P r with t ∈ T P, ++ T un and r ∈ T P . By Lemma 3.10 this is equivalent to requiring that π = π (cid:48) ◦ ψ t for some t ∈ T P, ++ T P un and π (cid:48) ∈ Irr( H P ). Definition 3.12.
A Langlands datum for H consists of a subset P ⊂ ∆ and anirreducible essentially tempered representation σ of H P in positive position.Equivalently, it can be given by a triple ( P, τ, t ), where P ⊂ ∆ , τ ∈ Irr( H P ) istempered and t ∈ T P, ++ T P, un . Then the associated H P -representation is σ = τ ◦ ψ t .The H -representations ind HH P ( σ ) and ind HH P ( τ ◦ ψ t ) are called standard.By Lemma 3.3 every standard H -module admits a central character.Now we can finally state the Langlands classification for affine Hecke algebras. Theorem 3.13. [Sol4, Theorem 2.4.4]
Let ( P, σ ) be a Langlands datum for H . ( a ) The H -representation ind HH P ( σ ) has a unique irreducible quotient, which we call L ( P, σ ) . ( b ) For every irreducible H -representation π there exists a Langlands datum ( P, σ ) with L ( P, σ ) ∼ = π . ( c ) If ( P (cid:48) , σ (cid:48) ) is another Langlands datum and L ( P (cid:48) , σ (cid:48) ) ∼ = L ( P, σ ) , then P (cid:48) = P and the H P -representations σ (cid:48) and σ are equivalent. Some consequences can be drawn immediately: • L ( P, σ ) is tempered if and only if P = ∆ and σ ∈ Irr( H ) is tempered (because L (∆ , σ ) = σ and Langlands data are unique). • In terms of Langlands data (
P, τ, t ) and ( P (cid:48) , τ (cid:48) , t (cid:48) ), the irreducible H -representations L ( P, τ, t ) = L ( P, τ ◦ ψ t ) and L ( P (cid:48) , τ (cid:48) , t (cid:48) ) = L ( P (cid:48) , τ (cid:48) ◦ ψ t (cid:48) )are equivalent if and only if P = P (cid:48) and τ ◦ t ∼ = τ (cid:48) ◦ t (cid:48) . (The latter does notimply t = t (cid:48) , for t and t (cid:48) may still differ by an element of T P ∩ T P .) • For q = 1, Corollary 2.1 shows that every standard module is irreducible.Hence the notions of irreducible representations and standard modules coin-cide for C [ X (cid:111) W ]. Example 3.14.
We work out the Langlands classification for H ( R , q ) with R oftype (cid:102) A . Its irreducible representations were already listed in paragraph 2.3. Theirreducible tempered representations are: • the Steinberg representation, • the parabolically induced representations ind H C [ X ] ( C t ) with t ∈ T un = S but t (cid:54) = − H C [ X ] ( C t − ) ∼ = ind H C [ X ] ( C t )), • the two representations π ( − , triv) and π ( − , St) which sum to ind H C [ X ] ( C − ). FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 29
These representations exhaust the Langlands data (
P, σ ) with P = ∆ = { α } . For P = ∅ we have H ∅ = C [ X ]. Its irreducible essentially tempered representations inpositive position are the C t with t ∈ T ++ T un = { z ∈ C × : | z | > } . We have • L ( ∅ , C t ) = ind H C [ X ] ( C t ) unless t = q , • L ( ∅ , C q ) = triv.In this way we obtain every element of Irr( H ( R , q ))–as listed at the end of paragraph2.3–exacly once, because T ++ T un is a fundamental domain for the action of W = (cid:104) s α (cid:105) on T ∼ = C × .Theorem 3.13 provides a quick and beautiful way to classify Irr( H ) in terms ofthe irreducible tempered representations of its parabolic subquotient algebras H P .On the downside, it conceals the topological structure of Irr( H ). For instance,with R of type (cid:102) A as discussed above, there is a family of H -representations π ( ∅ , triv , t ) = ind H C [ X ] ( C t ) with t ∈ T , irreducible for almost all t . The Langlandsclassification breaks it into two families, one with t ∈ T un and one with t ∈ T \ T un .In the next paragraph will see how this can be improved.We end this paragraph with some useful extras about the Langlands classification.The central character of an irreducible H P -representation τ is an element cc( τ ) of T P /W P . Its absolute value | cc( π ) | , with respect to the polar decomposition (3.6),lies in exp( a P ) /W P , and log | cc( π ) | ∈ a P /W P . We fix a W -invariant inner producton a . The norm of log | t | is the same for all representatives t ∈ T P of cc( τ ). Thatenables us to write (cid:107) cc( τ ) (cid:107) = (cid:107) log | t | (cid:107) for any t ∈ T P with W P t = cc( τ ) . Lemma 3.15. [Sol4, Lemma 2.2.6]
Let ( P, τ, t ) be a Langlands datum. ( a ) End H ( π ( P, τ, t )) = C id . ( b ) The representation L ( P, τ, t ) appears with multiplicity one in π ( P, τ, t ) = ind HH P ( τ ◦ ψ t ) . All other constituents L ( P (cid:48) , τ (cid:48) , t (cid:48) ) of π ( P, τ, t ) arelarger, in the sense that (cid:107) cc( τ (cid:48) ) (cid:107) > (cid:107) cc( τ ) (cid:107) . ( c ) Let
W s ∈ T /W . Both { π ∈ Irr( H ) : cc( π ) = W s } and { π ( P, τ, t ) : (
P, τ, t ) Langlands datum with cc( τ ) t ⊂ W s } are bases of the Grothendieck group of the category of finite dimensional H -module all whose O ( T ) -weights are in W s . With respect to a total ordering thatextends the partial ordering defined by part (b), the transition matrix betweenthese two bases is unipotent and upper triangular.
Discrete series representations.
In the representation theory of a reductive group G over a local field, Harish-Chandra showed that every irreducible tempered G -representation τ can be ob-tained from an irreducible square-integrable modulo centre representation δ of aLevi subgroup M of G , see [Kna, Theorem 8.5.3] and [Wal, Proposition III.4.1.i].More precisely, τ is a direct summand of the parabolic induction of δ . When thecentre of M is compact, δ is an isolated point of the space of irreducible tempered M -representations, and it is called a discrete series representation. Then it is asubrepresentation of L ( M ).Like for the Langlands classification, these results van be formulated and provenfor affine Hecke algebras as well. For these purposes it is essential that q s ∈ R > for all s ∈ S aff . To achieve that, we require not only that q ∈ R > (as we already did),but also that(3.8) λ ( α ) ∈ R , λ ∗ ( α ) ∈ R ∀ α ∈ R . This condition will be in force in the remainder of this paragraph.We note that the space a −− from (3.7) is empty unless R spans a ∗ , and in that case a −− = (cid:8) (cid:88) α ∈ ∆ x α α ∨ : x α < (cid:9) . For P ⊂ ∆ we have a P = Y P ⊗ Z R , t P = Y P ⊗ Z C , and that gives rise to a + P , a − P , T + P , T − P . Let a −− P = (cid:8) ν ∈ a P : (cid:104) δ, ν (cid:105) < ∀ δ ∈ a ∗ + P \ { } (cid:9) and T −− P = exp( a −− P ) be the versions of a −− and T −− = exp( a −− ) for R P . Definition 3.16.
Let ( π, V ) be a finite dimensional H -representation. We say that π belongs to the discrete series if the following equivalent conditions hold: • | x ( t ) | < t ∈ Wt( V ) and all x ∈ X + \ { } , • | Wt( V ) | ⊂ T −− , • Wt( V ) ⊂ T un T −− .Further, we can π essentially discrete series if the following equivalent conditionshold: • | x ( t ) | < t ∈ Wt( V ) and all x ∈ W aff X + \ { } , • | Wt( V ) | ⊂ T ∆ T −− , • Wt( V ) ⊂ T ∆ T un T −− .Here U T −− (for any U ⊂ T ) is considered as empty if T −− = ∅ . Example 3.17. • Let R be of type (cid:102) A and consider H = H ( R , λ, λ ∗ , q ) with λ ( α ) > λ ∗ ( α ) >
0. Here a −− = R < and T −− = (0 , q ( λ ( α )+ λ ∗ ( α )) / , so it is discrete se-ries. The trivial representation and all the twodimensional irreducible rep-resentation of H are not discrete series. From the classification in paragraph2.3, especially (2.9) and (2.10), we see that St is the only irreducible discreteseries H -representation if λ ( α ) = λ ∗ ( α ). When λ ( α ) > λ ∗ ( α ), π ( − , St) isthe only other irreducible discrete series representation, while π ( − , triv) isdiscrete series if λ ( α ) < λ ∗ ( α ). • The affine Hecke algebra H n ( q ) of type GL n has no discrete series, becauseits root datum R n is not semisimple. Here a −− ∆ = { ν ∈ R n : ν < , ν + ν < , . . . , ν + · · · + ν n − < , ν + · · · + ν n − + ν n = 0 } . For any z ∈ C × , the twist St ⊗ t z = St ◦ ψ t z of the Steinberg representation isessentially discrete series. In fact these are all irreducible essentially discreteseries representations of H n ( q ) (recall that q > • An affine Hecke algebra with λ = λ ∗ = 0 does not have discrete seriesrepresentations, apart from the case R = (0 , ∅ , , ∅ ), when the trivial repre-sentation of H = C is regarded as discrete series. FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 31
The relation between Definition 3.16 and representations of reductive p -adic groupsgoes via Casselman’s criterium for square-integrability, see [Cas, Theorem 4.4.6] and[Ren, § VII.1.2]. Opdam [Opd2, Lemma 2.22] translated this to various criteria for H -representations, which are equivalent with Definition 3.16.We condition (3.8) at hand, we can define a Hermitian inner product on H bydeclaring that { q ( w ) − / T w } is an orthonormal basis. Let L ( W ( R ) , q ) be theHilbert space completion of H with respect to this inner product–it is canonicallyisomorphic to L ( W ( R )). By [Opd2, Lemma 2.22], every irreducible discrete seriesrepresentation of H is isomorphic to a subrepresentation of the regular representa-tion of H on L ( W ( R ) , q ).In terms of representations of a reductive group G , ”essentially discrete series”means that a representation has finite length, and that its restriction to the de-rived group of G is square-integrable. An essentially discrete series representationis tempered if and only if { θ x : x ∈ X ∩ (∆ ∨ ) ⊥ } acts on it by characters from T ∆un . Theorem 3.18. [DeOp1, Theorem 3.22] and [Sol7, Lemma 1.3]( a ) Let π (cid:48) be an irreducible tempered H -representation. There exist P ⊂ ∆ and atempered essentially discrete series representation δ (cid:48) of H P such that π (cid:48) is adirect summand of ind HH P ( δ (cid:48) ) . ( b ) Let π be an irreducible essentially discrete series H P -representation. There exist t ∈ T P and a discrete series H P -representation δ such that π ∼ = δ ◦ ψ t . Clearly, a combination of Theorems 3.13 and 3.18 yields some description of Irr( H )in terms of parabolic induction and the discrete series of the subquotient algebras H P with P ⊂ ∆. We work this out in detail. Definition 3.19.
An induction datum for H is a triple ξ = ( P, δ, t ), where • P ⊂ ∆, • δ is an irreducible discrete series representation of H P , • t ∈ T P .We regard two triples ξ and ξ (cid:48) = ( P (cid:48) , δ (cid:48) , t (cid:48) ) as isomorphic (notation ξ ∼ = ξ (cid:48) ) if P = P (cid:48) , t = t (cid:48) and δ ∼ = δ (cid:48) . Let Ξ be the space of such induction data, topologized byregarding P and δ as discrete variables and T P as a complex analytic variety. Wesay that ξ = ( P, δ, t ) is positive, written ξ ∈ Ξ + , when | t | ∈ T P + .We already associated to such an induction datum the parabolically induced rep-resentation π ( P, δ, t ) = ind HH P ( δ ◦ ψ t ) . By Proposition 3.8(3.9) π ( P, δ, t ) is tempered ⇐⇒ t ∈ T P un . For ξ ∈ Ξ + we write P ( ξ ) = { α ∈ ∆ : | t ( α ) | = 1 } . Then P ⊂ P ( ξ ). This set of simple roots is useful because it allows to break theprocess of parabolic induction in two steps: the first dealing only with essentiallytempered representations and the second similar to the Langlands classification.More concretely, by Proposition 3.8 ind H P ( ξ ) ( H P ( ξ ) ) P ( δ ◦ ψ | t | ) is tempered, while π P ( ξ ) ( ξ ) := ind H P ( ξ ) H P ( δ ◦ ψ t ) is essentially tempered. Proposition 3.20. [Sol4, Proposition 3.1.4]
Let ξ = ( P, δ, t ) ∈ Ξ + and pick t P ( ξ ) ∈ T P ( ξ ) such that t P ( ξ ) t − ∈ T P ( ξ ) . ( a ) The H P ( ξ ) -representation π P ( ξ ) ( ξ ) is completely reducible and π P ( ξ ) ( ξ ) ◦ ψ − t P ( ξ ) is tempered. ( b ) Every irreducible summand of π P ( ξ ) ( ξ ) is of the form π P ( ξ ) ( P ( ξ ) , τ, t P ( ξ ) ) , where ( P ( ξ ) , τ, t P ( ξ ) ) is a Langlands datum for H . ( c ) The irreducible quotients of π ( ξ ) are the representations L ( P ( ξ ) , τ, t P ( ξ ) ) , with ( P ( ξ ) , τ, t P ( ξ ) ) coming from part (b). ( d ) Every irreducible H -representation is of the form described in part (c). ( e ) The functor
Ind HH P ( ξ ) induces an isomorphism End H P ( ξ ) (cid:0) π P ( ξ ) ( ξ ) (cid:1) ∼ −→ End H ( π ( ξ )) . For a given π ∈ Irr( H ), there are in general several induction data ξ ∈ Ξ + such that π is a quotient of π ( ξ ). So, in contrast with the Langlands classificaiton,Proposition 3.20 does not provide an actual parametrization of Irr( H ). To bring thatgoal closer, one has to analyse the relations between the various representations π ( ξ )with ξ ∈ Ξ.For u in the finite group T P ∩ T P = Hom Z (cid:0) X/ ( X ∩ Q P ) ⊕ ( X ∩ ( P ∨ ) ⊥ ) , C × (cid:1) , we have an automorphism ψ u of H P and a similar automorphism ψ P,u of H P , givenby ψ P,u ( θ x P T w ) = u ( x P ) θ x P T w x P ∈ X P , w ∈ W P . Then ( δ ◦ ψ − P,u ) ◦ ψ ut = δ ◦ ψ t , so(3.10) π ( P, δ ◦ ψ − P,u , ut ) = π ( P, δ, t ) . Suppose that w ∈ W and w ( P ) = P (cid:48) ⊂ ∆. Then(3.11) ψ w : H P → H P (cid:48) θ x T w (cid:48) (cid:55)→ θ w ( x ) T ww (cid:48) w − is an algebra isomorphism, and it descends to an algebra isomorphism ψ w : H P →H P (cid:48) . Moreover, ψ w can be implemented as conjugation by the element(3.12) ı ◦ w ∈ C ( X ) W ⊗ C [ X ] W H from Proposition 1.12, see [Sol4, (3.124)]. There is a bijection(3.13) I w : (cid:0) C ( X ) W ⊗ C [ X ] W H (cid:1) ⊗ H P V δ → (cid:0) C ( X ) W ⊗ C [ X ] W H (cid:1) ⊗ H P (cid:48) V δ h ⊗ v (cid:55)→ hı ◦ w − ⊗ v . By [Opd2, Theorem 4.33] it is isomorphism between that H -representationsind C ( X ) W ⊗ C [ X ] W HH P ( δ ◦ ψ t ) and ind C ( X ) W ⊗ C [ X ] W HH P (cid:48) ( δ ◦ ψ − w ◦ ψ w ( t ) ) . For t in a Zariski-open dense subset of T P , I w specializes to an H -isomorphism(3.14) π ( P, δ, t ) ∼ −→ π ( w ( P ) , δ ◦ ψ − w , w ( t )) . As such, I w : H ⊗ H P V δ → H ⊗ H P (cid:48) V δ is a rational map in the variable t ∈ T P (possibly with poles for some t ∈ T P ). FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 33
Lemma 3.21.
For all t ∈ T P and all w ∈ W with w ( P ) ⊂ ∆ , the representations π ( P, δ, t ) and π ( w ( P ) , δ ◦ ψ − w , w ( t )) have the same irreducible constituents, with thesame multiplicities.Proof. The above implies that, for t in a Zariski-open subset of T P , these two H -representations have the same character. The vector space H ⊗ H P V δ does notdepend on t and the character of π ( P, δ, t ) depends algebraically on t ∈ T P , so infact π ( P, δ, t ) and π ( w ( P ) , δ ◦ ψ − w , w ( t )) have the same character for all t ∈ T P .Since H is of finite rank as module over its centre, the Frobenius–Schur theorem[CuRe, Theorem 27.8] applies, and says that the characters of inequivalent irre-ducible H -representation are linearly independent functionals on H . As the char-acter of π ( P, δ, t ) determines that representation up to semisimplification, it carriesenough information to determines the multiplicities with which the irreducible rep-resentations appear in π ( P, δ, t ). (cid:3) The next results are much deeper, for their proofs involve a study of topologicalcompletions of affine Hecke algebras [Opd2, DeOp1]. For a systematic bookkeepingof the H -isomorphisms (3.10) and (3.14), we put them in a groupoid W Ξ . Its basespace is the power set of ∆, the collection of morphisms from P to P (cid:48) is W Ξ ,P P (cid:48) = { ( w, u ) ∈ W × ( T P ∩ T P ) : w ( P ) = P (cid:48) } , and the composition comes from the group T (cid:111) W . This groupoid acts on the spaceof induction data Ξ as( w, u )( P, δ, t ) = w · ( P, δ ◦ ψ − u , ut ) = ( w ( P ) , δ ◦ ψ − u ◦ ψ − w , w ( ut )) . Theorem 3.22.
Let ξ = ( P, δ, t ) , ξ (cid:48) = ( P (cid:48) , δ (cid:48) , t (cid:48) ) ∈ Ξ + . The H -representations π ( ξ ) and π ( ξ (cid:48) ) have a common irreducible quotient if and only if there exists a ( w, u ) ∈ W Ξ with ( w, u ) ξ = ξ (cid:48) . When t, t (cid:48) ∈ T un , Proposition 3.20.a says that π ( ξ ) and π ( ξ (cid:48) ) are completelyreducible. Then the statement of the theorem becomes: π ( ξ ) and π ( ξ (cid:48) ) have acommon irreducible subquotient if and only if ξ (cid:48) ∈ W Ξ ξ . This is an analogue ofLanglands’ disjointness theorem (see [Kna, Theorem 14.90] for real reductive groupsand [Wal, Proposition 4.1.ii] for p -adic reductive groups), and it was proven in[DeOp1, Corollary 5.6]. The generalization to ξ, ξ (cid:48) ∈ Ξ + was established in [Sol4,Theorem 3.3.1.a].Every element ( w, u ) ∈ W Ξ gives rise to an intertwining operator π (( w, u ) , P, δ, t ) : π ( P, δ, t ) → π ( w ( P ) , δ ◦ ψ − u ◦ ψ − w , w ( ut )) . It is rational as a function of t ∈ T P and regular for almost all t ∈ T P . Namely,the isomorphisms (3.14) come from (3.13), while (3.10) is just the identity on theunderlying vector space. For isomorphic induction data ( P, δ, t ) and (
P, σ, t ) we alsoneed an H -isomorphism ind HH P ( δ ◦ ψ t ) → ind H P H ( σ ◦ ψ t ) . To that end we pick (independently of t ) an H P -isomorphism δ ∼ = σ (which anywayis unique up to scalars) and apply ind HH P to that. In this way we get an intertwiningoperator π (( w, u ) , ξ ) : π ( ξ ) → π ( ξ (cid:48) ) whenever ( w, u ) ξ and ξ (cid:48) are isomorphic. This operator is unique up to scalars and(3.15) π (( w (cid:48) , u (cid:48) ) , ( w, u ) ξ ) ◦ π (( w, u ) , ξ ) = (cid:92) (( w (cid:48) , u (cid:48) ) , ( w, u )) π (( w (cid:48) , u (cid:48) )( w, u ) , ξ )for some (cid:92) (( w (cid:48) , u (cid:48) ) , ( w, u )) ∈ C × . Theorem 3.23.
Let ξ, ξ (cid:48) ∈ Ξ + . The operators { π (( w, u ) , ξ ) : ( w, u ) ∈ W Ξ , ( w, u ) ξ ∼ = ξ (cid:48) } are regular and invertible, and they span Hom H ( π ( ξ ) , π ( ξ (cid:48) )) . In case the coordinates t, t (cid:48) of ξ, ξ (cid:48) lie in T un , this is shown in [DeOp1, Corollary5.4]. That version is analogue of Harish-Chandra’s completeness theorem, see [Kna,Theorem 14.31] and [Sil, Theorem 5.5.3.2]. For the version with arbitrary ξ, ξ (cid:48) ∈ Ξ + we refer to [Sol4, Theorem 3.3.1.b].The relations between parabolically induced representations π ( ξ ) and π ( ξ (cid:48) ) with ξ, ξ (cid:48) ∈ Ξ W Ξ -associate but not positive are more complicated, and not understoodwell. For the principal series π ( ∅ , triv , t ) this issue was investigated in [Ree1].Finally, everything is in place to formulate an extension of the Langlands classifi-cation that incorporates results about discrete series representations. The outcomeis similar to L-packets in the local Langlands correspondence. Recall that (3.8) isin force. Theorem 3.24. [Sol4, Theorem 3.3.2]
Let π be an irreducible H -representation. There exists a unique W Ξ -association class ( P, δ, t ) ∈ Ξ / W Ξ such that the following equivalent statements hold: ( a ) π is isomorphic to an irreducible quotient of π ( ξ + ) , for some ξ + ∈ Ξ + ∩W Ξ ( P, δ, t ) ; ( b ) π is a constituent of π ( P, δ, t ) , and (cid:107) cc ( δ ) (cid:107) is maximal for this property.Further, π is tempered if and only if t ∈ T un . Theorem 3.24 associates to every irreducible H -representation π an essentiallyunique positive induction datum ( P, δ, t ). If ξ (cid:48) = ( P (cid:48) , δ (cid:48) , t (cid:48) ) is another positiveinduction datum associated to π , then ξ (cid:48) W Ξ -associate to ( P, δ, t ) and Theorem3.23 entails that π ( ξ (cid:48) ) ∼ = π ( P, δ, t ). Hence the parabolically induced representation π ( P, δ, t ) is uniquely determined by π , up to isomorphism of H -representations.Conversely, however, π ( P, δ, t ) can have more than one irreducible quotient. So,like an L-packet for a reductive group can have more than one element, (
P, δ, t )might be associated (in Theorem 3.24) to several irreducible H -representations. Example 3.25. • Consider R of type (cid:102) A , H = H ( R , q ). Then W Ξ , ∅∅ = { , s α } , W Ξ , ∆∆ = { } and Theorem 3.24 works out as follows, where we take t or t − depending on whether | t | ≥ | t | ≤ H ) induction dataSt (cid:55)→ (∆ = { α } , St , (cid:55)→ ( ∅ , triv , q ) π ( − , St) , π ( − , triv) (cid:55)→ ( ∅ , triv , − H C [ X ] ( C t ) (cid:55)→ ( ∅ , triv , t ± ) FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 35 • Keep R of type (cid:102) A , but consider H = H ( R ,
1) = C [ W aff ]. In this case H does not have any discrete series representations, and the effect of Theorem3.24 is: Irr( H ) induction dataSt , triv (cid:55)→ ( ∅ , triv , π ( − , St) , π ( − , triv) (cid:55)→ ( ∅ , triv , − H C [ X ] ( C t ) (cid:55)→ ( ∅ , triv , t ± ) • For R of type GL n , H = H n ( q ), and P ⊂ ∆ given by a partition (cid:126)n =( n , n , . . . , n d ) of n , we have W Ξ ,P P / ( T P ∩ T P ) = (cid:89) m ≥ S ( { i : n i = m } ) ∼ = N S n (cid:16) d (cid:89) i =1 S n i (cid:17)(cid:46) d (cid:89) i =1 S n i . The only irreducible discrete series representations of H ( R n,P , q ) are St ⊗ t z = St ◦ ψ P,t z with t z ∈ T P ∩ T P . Recall that T P ∼ = (cid:89) di =1 (( C × ) n i ) S ni ∼ = ( C × ) d . An induction datum ( P, St ⊗ t z , (cid:126)z ) with (cid:126)z = ( z , . . . , z d ) ∈ T P is positive ifand only if | z | ≥ | z | · · · ≥ | z n | , a condition which is preserved by the actionof T P ∩ T P on such induction data. Hence, up W Ξ -association, it suffices toconsider only positive induction data ( P, St , t )–so with t z = 1.It is easily checked that every pair ( (cid:126)n, (cid:126)z ) as in Theorem 2.6.b is S n -associate to a pair ( (cid:126)n, (cid:126)z ) which is positive in the above sense, and that thelatter is unique up to W Ξ -association. The irreducible H n ( q )-representationattached to ( (cid:126)n, (cid:126)z ) in paragraph 2.4 is a quotient of π ( P, St , (cid:126)z ) = π ( (cid:126)n, (cid:126)z ), justas in Theorem 3.24. Thus, for H n ( q ) Theorem 3.24 recovers Theorem 2.6:both parametrize Irr( H n ( q )) bijectively with basically the same data.3.4. Lusztig’s reduction theorems.
In Paragraph 1.5 we already hinted at a link between affine Hecke algebras andgraded Hecke algebras. The connected is established with two reduction steps, whoseoriginal versions are due to Lusztig [Lus3]. These simplifications do not work inthe same way for all representations, they depend on the central characters. Thefirst reduction step limits the set of central characters that has to be considered tounderstand all finite length H -modules.By [Lus3, Lemma 3.15], for t ∈ T and α ∈ R :(3.16) s α ( t ) = t ⇐⇒ α ( t ) = (cid:26) α ∨ / ∈ Y ± α ∨ ∈ Y .
Fix t ∈ T . We will exhibit an algebra, almost a subalgebra of H , that capturesthe behaviour of H -representations with central character close to W t ∈ T /W . Weconsider the root system R (cid:48) t = { α ∈ R : s α ( t ) = t } . It fits in a root datum R (cid:48) t = ( X, R t , Y, R ∨ t ), and λ and λ ∗ restrict to parameterfunctions for R (cid:48) t . The affine Hecke algebra H ( R (cid:48) t , λ, λ ∗ , q ) naturally embeds in H = H ( R , λ, λ ∗ , q ).But we have to be careful, this construction is not suitable when c α ( t ) for some α ∈ R . For instance, when R is of type (cid:102) A and t = q (cid:54) = 1, the algebra H ( R (cid:48) t , λ, λ ∗ , q ) is just C [ X ]. That is hardly helpful to describe all H -modules with weights in W q (e.g. ind H C [ X ] ( C q ), triv and St).When c α ( t ) (cid:54) = 0 for all α ∈ R , H ( R (cid:48) t , λ, λ ∗ , q ) detects most of what can happenwith H -representations with central character W t , but still not everything.
Example 3.26.
Consider the root datum R of type (cid:102) A , with X = { x ∈ Z : x + x + x = 0 } , R = { e i − e j : i (cid:54) = j } ∼ = A , ∆ = { e − e , e − e } . The point t ∈ T with t ( e − e ) = t ( e − e ) = e πi/ satisfies R t = ∅ but W t = { id , (123) , (132) } . We have H ( R t , λ, λ ∗ , q ) = C [ X ], which possesses onlyone irreducible representation with central character t . On the other hand, the R-group for H and ξ = ( ∅ , triv , t ) is W t . Theorem 4.2 entails that ind H C [ X ] ( C t ) splitsinto three inequivalent irreducible H -representations, all with central character W t .The set(3.17) R t = { α ∈ R : s α ( t ) = t, c α ( t ) (cid:54) = 0 } is a root system, because λ and λ ∗ are W -invariant. When t ∈ T un and (3.8) holds, R t coincides with R (cid:48) t because c α ( t ) cannot be 0.Let R + t = R + ∩ R t be the set of positive roots, and let ∆ t be the unique basis of R t contained in R + t . We warn that ∆ t need not be a subset of ∆. The groupΓ t = { w ∈ W t : w ( R + t ) = R + t } , satisfies W t = W ( R t ) (cid:111) Γ t . For every w ∈ Γ t there exists an algebra automorphismlike (3.11) ψ w : H ( R t , λ, λ ∗ , q ) → H ( R t , λ, λ ∗ , q ) θ x T w (cid:48) (cid:55)→ θ w ( x ) T ww (cid:48) w − . This is a group action of Γ t , so we can form the crossed product H ( R t , λ, λ ∗ , q ) (cid:111) Γ t .We recall that this means H ( R t , λ, λ ∗ , q ) ⊗ C [Γ t ] as vector spaces, with multiplicationrule ( h ⊗ γ )( h (cid:48) ⊗ γ (cid:48) ) = hψ γ ( h (cid:48) ) ⊗ γγ (cid:48) . The group Γ t can be embedded in (cid:0) C ( X ) W ⊗ C [ X ] W H ( R , λ, λ ∗ , q ) (cid:1) × with the elements ı ◦ w from Proposition 1.12. In view of (3.11) and (3.12), this realizes H ( R t , λ, λ ∗ , q ) (cid:111) Γ t as a subalgebra of C ( X ) W ⊗ C [ X ] W H ( R , λ, λ ∗ , q ).Unfortunately tensoring with C ( X ) W kills many interesting representations, sothe above does not yet suffice to relate the module categories of H ( R , λ, λ ∗ , q ), of H ( R t , λ, λ ∗ , q ) (cid:111) Γ t and of a graded Hecke algebra. To achieve that, some techni-calities are needed.For an (analytically) open W -stable subset U of T , let C an ( U ) be the algebra ofcomplex analytic functions on U . As Z ( H ) injects in C an ( U ) W , we can form thealgebra H an ( U ) := C an ( U ) W ⊗ C [ X ] W H ( R , λ, λ ∗ , q ) . It can also be obtained from Definition 1.11 by using C an ( U ) instead of O ( T ), andas a vector space it is just C an ( U ) ⊗ C H ( W, q ).Let Mod f,U be the category of finite length H -modules, all whose O ( T )-weightslie in U . Proposition 3.27.
The inclusion
H → H an ( U ) induces an equivalence of categories Mod f ( H an ( U )) → Mod f,U ( H ) . FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 37
Let U t be a ”sufficiently small” W t -invariant open neighborhood of t in T and put U = W U t . Then C an ( U ) = (cid:77) w ∈ W/W t C an ( wU t )and there is a well-defined group homomorphismΓ t → H an ( U ) × : w (cid:55)→ U t ı ◦ w . This realizes H ( R t , λ, λ ∗ , q ) (cid:111) Γ t as a subalgebra of H an ( U ). Our version of Lusztig’sfirst reduction theorem [Lus3, §
8] is:
Theorem 3.28.
Assume that (3.8) holds and that t ∈ T satisfies W | t | − t = W t .Then there exist open W t -stable neighborhoods U t of t with the following properties. ( a ) There is a natural embedding of C an ( U ) W -algebras H ( R t , λ, λ ∗ , q ) an ( U t ) (cid:111) Γ t → H an ( U ) . ( b ) Part (a) and Proposition 3.27 induce equivalences of categories
Mod f,U t ( H ( R t , λ, λ ∗ , q ) (cid:111) Γ t ) ∼ = Mod f ( H ( R t , λ, λ ∗ , q ) an ( U t ) (cid:111) Γ t ) ∼ = Mod f ( H an ( W U t )) ∼ = Mod f,W U t ( H ) . ( c ) When t ∈ T un , we may take U t of the form U (cid:48) t × exp( a ) , where U (cid:48) t ⊂ T un is asmall open W t -stable ball around t . In that case the equivalences of categoriesbetween the outer terms in part (b) preserve temperedness and (essentially) dis-crete series. Parts (a) and (b) of Theorem 3.28 can be found in [BaMo2, Theorem 3.3] and[Sol4, Theorem 2.1.2]. For part (c) we refer to [AMS3, Proposition 2.7].The conditions in Theorem 3.28 avoid possible unpleasantness caused by non-invertible intertwining operators. Algebras of the form H ( R t , λ, λ ∗ , q ) (cid:111) Γ t behavealmost the same as affine Hecke algebras. The difference can be handled with Cliffordtheory, as in [RaRa, Appendix]. In fact everything we said so far in this paper canbe generalized to such crossed product algebras, see [Sol4, AMS3]. For simplicity,we prefer to keep the finite groups Γ t out of our presentation.An advantage of Theorem 3.28 is that it reduces the study of Mod f ( H ) to thosemodules of H ( R t , λ, λ ∗ , q ) (cid:111) Γ t all whose C [ X ]-weights belong to a small neighbor-hood of the point t , which is fixed by W ( R t ) (cid:111) Γ t . We will use this in a loose sense,suppressing Γ t . Then Theorem 3.28 says that it suffices to consider those finitedimensional modules of an affine Hecke algebra H , all whose C [ X ]-weights lie in asmall neighborhood of a W -fixed point t ∈ T .The second reduction theorem will transfer such H -representations to representa-tions of graded Hecke algebras. In the remainder of this paragraph we assume that u is fixed by W . We define a parameter function k u for the root system R u by(3.18) k u ( α ) = ( λ ( α ) + α ( u ) λ ∗ ( α )) log( q ) / . By (3.16) α ( u ) ∈ {± } , so k u is real-valued whenever the positivity condition (3.8)for q holds. This gives a graded Hecke algebra H ( t , W, k u ).Let V ⊂ t be an analytically open W -stable subset. We can form the algebra H ( t , W, k u ) an ( V ) = C an ( V ) W ⊗ O ( t ) W H ( t , W, k u ) , which as vector space is just C an ( V ) ⊗ O ( t ) H ( t , W, k u ) = C an ( V ) ⊗ C C [ W ] . Let Mod f,V ( H ( t , W, k u )) be the category of those finite dimensional H ( t , W, k u )-modules, all whose O ( t )-weights lie in V . An analogue of Proposition 3.27 says thatthe inclusion H ( t , W, k u ) → H ( t , W, k u ) an ( V )induces an equivalence of categories(3.19) Mod f ( H ( t , W, k u ) an ( V )) ∼ −→ Mod f,V ( H ( t , W, k u )) . The analytic map exp u : t → T, λ (cid:55)→ u exp( λ )is W -equivariant, because u is fixed by W . It gives rise to algebra homomorphismsexp ∗ u : C an (exp u ( V )) → C an ( V ) f (cid:55)→ f ◦ exp u Φ u : C ( X ) W ⊗ C [ X ] W H an (exp u ( V )) → Q ( S ( t ∗ )) W ⊗ S ( t ∗ ) W H ( t , W, k u ) an ( V ) f ı ◦ w (cid:55)→ ( f ◦ exp u )˜ ı w . In good circumstances, the works already without involving rational (non-regular)functions.
Theorem 3.29. [Lus3, Theorem 9.3], [BaMo2, §
4] and [Sol4, Theorem 2.1.4]
Let V ⊂ t be an (analytically) open subset such that • V is W -stable, • exp u is injective on V , • for all α ∈ R, λ ∈ V the numbers (cid:104) α, λ (cid:105) , (cid:104) α, λ (cid:105) + k u ( α ) do not lie in πi Z \{ } . ( a ) exp ∗ u : C an (exp u ( V )) → C an ( V ) is a W -equivariant algebra isomorphism, andmakes H an (exp u ( V )) into a C an ( V ) W -algebra. ( b ) Φ u restricts to an isomorphism of C an ( V ) W -algebras H an (exp u ( V )) → H ( t , W, k u ) an ( V ) . Notice that the conditions in Theorem 3.29 always hold when V is a small neigh-borhood of 0 in t . When k u is real-valued, for instance whenever (3.8) holds, thesecondition also hold for V of the form a + V (cid:48) with V (cid:48) ⊂ i a a small ball around 0.Combining Theorems 3.28, 3.29 and Proposition 3.27, we can draw importantconsequences for the category of finite dimensional H -modules Corollary 3.30. [Sol4, Corollary 2.15]
Assume that (3.8) holds and that u ∈ T un . ( a ) For λ ∈ a the categories Mod f,W u exp( λ ) ( H ) and Mod f,W u λ ( H ( t , W ( R u ) , k u ) (cid:111) Γ u ) are naturally equivalent. ( b ) The categories
Mod f,W u exp( a ) ( H ) and Mod f, a ( H ( t , W ( R u ) , k u ) (cid:111) Γ u ) are natu-rally equivalent. ( c ) The equivalences of categories from parts (a) and (b) are compatible with para-bolic induction.
Analogues for graded Hecke algebras.
Motivated by Corollary 3.30 we investigate all finite dimensional representationsof graded Hecke algebras. The representation theory of graded Hecke algebras hasbeen developed together with that of affine Hecke algebras, and they are very similar.As far as the topics in this section are concerned, these two kinds of algebras behaveanalogously. Instead of translating the paragraphs 3.1–4.1 to graded Hecke algebras,
FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 39 we will be more sketchy here, just providing the necessary definitions and pointingout the analogies. For more background and proofs we refer to [BaMo2, Eve, KrRa,Sol2, Sol3].Compared to Paragraph 1.5, we assume in addition that W is a crystallographicWeyl group. Equivalently, the data t , a , R, W, S for a graded Hecke algebra now comefrom a based root datum R = ( X, R, Y, R ∨ , ∆). Let k : R → C be a W -invariantparameter function and consider the algebra H = H ( t , W, k ).A parabolic subalgebra of H is by definition of the form H P = H ( t , W P , k ) for asubset P ⊂ ∆. The associated ”semisimple” quotient algebra is H P = H ( t P , W P , k ).The relation between these two subquotients of H is simple: H P = H P ⊗ C S ( t P ∗ ) = H P ⊗ C O ( t P ) . In particular every irreducible representation of H P is of the form π P ⊗ λ for unique π P ∈ Irr( H P ) and λ ∈ t P . Parabolic induction for H is the functor ind H H P . Lemma 3.31. [AMS3, Theorems 2.5.b and 2.11.b]
The equivalences of categories from Theorems 3.28 and 3.29 and Corollary 3.30 arecompatible with parabolic induction.
Finite dimensional H -modules have weights with respect to the commutative sub-algebra S ( t ∗ ) = O ( t ). With Theorem 3.29 one can translate many notions for H -modules to H -modules. In terms of weights, this boils down to replacing a sub-set U ⊂ T by exp − ( U ) ⊂ t . More concretely, we say that a finite dimensional H -representation π is • tempered if Wt( π ) ⊂ i a + a − , • essentially tempered if Wt( π ) ⊂ t ∆ + i a + a − , • discrete series if Wt( π ) ⊂ i a + a −− , • essentially discrete series if Wt( π ) ⊂ t ∆ + i a + a −− .Here V + a −− (for any V ⊂ t ) is considered as empty when a −− = ∅ .Then a Langlands datum for H is a triple ( P, τ, λ ), where P ⊂ ∆, τ ∈ Irr( H P ) istempered and λ ∈ i a P + a P ++ . Then ind HH P ( τ ⊗ λ ) is called a standard H -module.With these definitions, the Langlands classification (as in Theorem 3.13 and Lemma3.15) holds true for graded Hecke algebras [Eve].As expected, and proven in [AMS3, Theorem 2.11.d], the equivalence of categoriesbetween Mod f,V ( H ( t , W, k u )) and Mod f, exp u ( V ) ( H ) resulting from Theorem 3.29 andProposition 3.27 preserves temperedness and (essentially) discrete series. Combiningthat with Theorem 3.28, we find: Lemma 3.32.
Assume that (3.8) holds. The equivalence of categories between
Mod f,W u exp( a ) ( H ) and Mod f, a ( H ( t , W ( R u ) , k u ) (cid:111) Γ u ) from Corollary 3.30 preservestemperedness and (essentially) discrete series. From now on we assume that k : R → C has values in R , so that H is related toan affine Hecke algebra with parameters in R > . As induction data for H we taketriples ˜ ξ = ( P, δ, λ ) where P ⊂ ∆, ( δ, V δ ) ∈ Irr( H P ) is discrete series and λ ∈ t P .The space of such triples (with δ considered up to isomorphism) is denoted ˜Ξ. Theparabolically induced representation attached to an induction datum is π ( P, δ, λ ) = ind HH P ( δ ⊗ λ ) . Lemma 3.33.
Let ˜ ξ = ( P, δ, λ ) ∈ ˜Ξ . ( a ) π ( ˜ ξ ) is tempered if and only if λ ∈ i a P . In that case π ( ˜ ξ ) is completely reducible. ( b ) Let W P cc( δ ) be the central character of δ . The central character of π ( ˜ ξ ) is W (cc( δ ) + λ ) . It lies in a /W if and only if λ ∈ a P .Proof. The arguments from these statement will be given in the setting of affineHecke algebras. From there they can be translated to graded Hecke algebras withParagraph 3.4.(a) follows from Propositions 3.8 and 3.20.a.(b) The expression for the central character comes from Lemma 3.3. Since k is real-valued and δ is discrete series, W P cc( δ ) lies in a P /W P [Slo2, Lemma 2.13]. Thesetwo facts imply the second statement. (cid:3) The collection of intertwining operators between the representation π ( ˜ ξ ) with˜ ξ ∈ ˜Ξ is simpler than for affine Hecke algebras, because t P ∩ t P = { } does notcontribute to it. There is a groupoid W ˜Ξ over ∆, with W ˜Ξ ,P P (cid:48) = { w ∈ W : w ( P ) = P (cid:48) } . To every w ∈ W ˜Ξ ,P P (cid:48) one can associate an algebra isomorphism ψ w : H P → H w ( P ) x ⊗ w (cid:48) (cid:55)→ w ( x ) ⊗ ww (cid:48) w − x ∈ t ∗ , w (cid:48) ∈ W P . Then w ( ˜ ξ ) = ( w ( P ) , δ ◦ ψ − w , w ( λ )) is another induction datum, and there is anintertwining operator I w : π ( P, δ, λ ) → π ( w ( P ) , δ ◦ ψ − w , w ( λ )) . The latter is rational as a function of λ ∈ t P and comes from (cid:0) Q ( S ( t ∗ )) W ⊗ S ( t ∗ ) W H (cid:1) ⊗ H P V δ → (cid:0) Q ( S ( t ∗ )) W ⊗ S ( t ∗ ) W H (cid:1) ⊗ H w ( P ) V δ h ⊗ v (cid:55)→ h ˜ ı w ⊗ v , where ˜ ı w is as in Proposition 1.14. We call an induction datum ˜ ξ = ( P, δ, λ ) positiveif λ ∈ i a P + a P + , and we define P ( ˜ ξ ) = { α ∈ ∆ : (cid:60)(cid:104) α, λ (cid:105) = 0 } . Using these notions, the whole of paragraph 3.3 holds for graded Hecke algebras.see [Sol3].
Example 3.34.
Consider a = a ∗ = R , t = t ∗ = C , R = {± } , ∆ = { α = 1 } , W = (cid:104) s α (cid:105) . The graded Hecke algebra H = H ( t , W, k ) with k ( α ) = k > | O ( t ) = C − k and St | C [ W ] = sign.Apart from ( ∅ , St , ∅ , triv , λ ) with λ ∈ i a + a + = i R + R ≥ . For λ (cid:54) = k , π ( ∅ , triv , λ ) = ind H O ( t ) is irreducible, while π ( ∅ , triv , k ) has the”trivial representation” as unique irreducible quotient. It is given by triv | O ( t ) = C k and triv | C [ W ] = triv.4. Classification of irreducible representations
As in paragraph 3.3, we work with an affine Hecke algebra H = H ( R , λ, λ ∗ , q )where q > λ, λ ∗ are real-valued. We abbreviate H = H ( t , W, k ) where W = W ( R ) , t = Lie( T ) and k is a real-valued parameter function. FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 41
In Theorem 3.24 we reduced the classification of irreducible H -representations toa little combinatorics with a groupoid W Ξ and two substantial subproblems: • classify the irreducible discrete series representations δ of the parabolic sub-quotient algebras H P (modulo the action of T P ∩ T P via the automorphisms ψ P,u ), • determine the irreducible quotients of π ( ξ ) for ξ = ( P, δ, t ) ∈ Ξ + .In this section we address both these issues.4.1. Analytic R-groups.
By Proposition 3.20.a–c the second subproblem above is equivalent to classifyingthe irreducible summands of the completely reducible H P ( ξ ) -representation π P ( ξ ) ( ξ ) := ind H P ( ξ ) H P ( δ ◦ ψ t ) . For that we have to analyse End H P ( ξ ) ( π P ( ξ ) ( ξ ), which by Proposition 3.20.e boilsdown to investigating End H ( π ( ξ )).For ξ = ( P, δ, t ) ∈ Ξ + we let W ξ be the subgroup of W Ξ ,P P that stabilizes ξ (up toisomorphism of induction data). From Theorem 3.23 we know that the intertwiningoperators π ( w, ξ ) with w ∈ W ξ span End H ( π ( ξ )), but they need not be linearlyindependent. Knapp and Stein exhibited a subgroup R ξ of W ξ such that the π ( w, ξ )with w ∈ R ξ do form a basis of End H ( π ( ξ )).Let R + P be the set of positive roots in R P , with respect to the basis P . Supposethat α ∈ R + \ R + P and that P ∪ { α } is a basis of a parabolic root subsystem R P ∪{ α } of R . Then we put α P = α | a P ∗ and c Pα = (cid:89) β ∈ R + P ∪{ α } c β ∈ C ( X ) . We note that c Pα is W P -invariant because W P stabilizes R P and does not makepositive roots outside R P negative. Let δ ∈ Irr( H P ) be discrete series, with centralcharacter cc( δ ) = W P r . Then t (cid:55)→ c Pα ( rt ) is a rational function on T P , independentof the choice of the representative r for cc( δ ). For ξ = ( P, δ, t ) we consider thefollowing subset of a P ∗ : R ξ = (cid:8) ± α P : α ∈ R + \ R + P as above, c Pα has a non-removable pole at rt (cid:9) . This generalizes the root system R t from (3.17). Proposition 4.1. [DeOp2, Proposition 4.5]( a ) R ξ is a reduced root system in a P ∗ , ( b ) the Weyl group W ( R ξ ) is naturally a normal subgroup of W ξ . The group W ξ acts naturally on a P ∗ and stabilizes all the data used to construct R ξ , so it also acts naturally on R ξ . Clearly R + determines a set of positive roots R + ξ in R ξ . We define(4.1) R ξ = { w ∈ W ξ : w ( R + ξ ) = R + ξ } . As W ( R ξ ) acts simply transitively on the collection of positive systems of R ξ :(4.2) W ξ = W ( R ξ ) (cid:111) R ξ . Theorem 4.2.
Let ξ ∈ Ξ + . ( a ) For w ∈ W ξ , the intertwining operator π ( w, ξ ) is scalar if and only if w ∈ W ( R ξ ) . ( b ) There exists a 2-cocycle (cid:92) ξ : R ξ × R ξ → C × (depending on the normalization ofthe operators π ( w, ξ ) with w ∈ W ξ ) such that End H ( π ( ξ )) = span { π ( w, ξ ) : w ∈ R ξ } is isomorphic to the twisted group algebra C [ R ξ , (cid:92) ξ ] . The multiplication in C [ R ξ , (cid:92) ξ ] is as in (3.15) . ( c ) Given the normalization of these intertwining operators, we write π P ( ξ ) ( ξ, ρ ) = Hom C [ R ξ ,(cid:92) ξ ] (cid:0) ρ, π P ( ξ ) ( ξ ) (cid:1) . There are bijections
Irr( C [ R ξ , (cid:92) ξ ]) → { irreducible summands of → { irreducible quotients of π P ( ξ ) ( ξ ) , up to isomorphism } π ( ξ ) , up to isomorphism } ρ (cid:55)→ π P ( ξ ) ( ξ, ρ ) (cid:55)→ L (cid:0) P ( ξ ) , π P ( ξ ) ( ξ, ρ ) (cid:1) Proof.
For ξ = ( P, δ, t ) with t ∈ T P un , all this (and more) was shown in [DeOp2, The-orems 5.4 and 5.5]. Using Proposition 3.20.a, the same proofs work for π P ( ξ ) ( P, δ, t )with any ξ ∈ Ξ, they show the theorem on the level of H P ( ξ ) . Finally, we applyProposition 3.20.c,e. (cid:3) Example 4.3. • R of type (cid:102) A , H = H ( R , q ). The root system R ξ is nonemptyonly for ξ = ( ∅ , triv , R ξ = R, W ξ = W = W ( R ξ ) , R ξ = 1and π ( ξ ) = ind H C [ X ] ( C ) is irreducible.The R-group R ξ is nontrivial only for ξ = ( ∅ , triv , − R ξ = ∅ , W ξ = R ξ = W . Further π ( ξ ) = ind H C [ X ] ( C − ) is reducible andEnd H ( π ( ξ )) ∼ = C [ R ξ ]. With the appropriate normalization of the intertwin-ing operator π ( s α , ξ ), we have π ( ξ, ρ = triv) = π ( − , triv) , π ( ξ, ρ = sign) = π ( − , St) . • R = R n , H = H n ( q ). For all ξ = ( P, δ, t ) ∈ Ξ, W ξ = { w ∈ S n : w ( P ) = P, w ( t ) = t } = W ( R ξ )and R ξ = 1. HenceEnd H P ( ξ ) ( π P ( ξ ) ( ξ )) ∼ = End H ( π ( ξ )) ∼ = C and π ( ξ ) has only one irreducible quotient (as we already saw in severalways). • R arbitrary, λ = λ ∗ = 0 , H = H ( R ,
1) = C [ X (cid:111) W ]. The only discrete seriesrepresentation of a parabolic subquotient algebra of this H is the trivialrepresentation of H ∅ = C . HenceΞ = { ( ∅ , triv , t ) : t ∈ T } . Further c α = 1 for all α ∈ R , so R ξ is empty for all ξ ∈ Ξ. As T ∅ ∩ T ∅ = T ∅ = { } , W ξ = W ( ∅ , triv ,t ) = W t = R ξ . Here End H (ind H C [ X ] ( C t )) ∼ = C [ W t ] acts on the vector space ind H C [ X ] ( C t ) ∼ = C [ W ] as the induction, from W t to W of the right regular representation of W t , as can be inferred from (3.13). FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 43
The last example shows that R-groups can be as complicated as W itself. Thisis in sharp contrast with the situations for real reductive groups and for classical p -adic groups, where all R-groups are abelian 2-groups. In all examples that we areaware of, the 2-cocycle (cid:92) ξ of R ξ is a coboundary, so that C [ R ξ , (cid:92) ξ ] is isomorphic to C [ R ξ ]. It would be interesting to know whether or not this is always true for affineHecke algebras.By Proposition 3.20.b, the H -module(4.3) ind HH P ( ξ ) ( π P ( ξ ) ( ξ, ρ )) = ind HH P ( ξ ) (cid:0) Hom C [ R ξ ,(cid:92) ξ ] ( ρ, π P ( ξ ) ( ξ )) (cid:1) =Hom C [ R ξ ,(cid:92) ξ ] (cid:0) ρ, ind HH P ( ξ ) π P ( ξ ) ( ξ ) (cid:1) = Hom C [ R ξ ,(cid:92) ξ ] (cid:0) ρ, π ( ξ )) = π ( ξ, ρ )from Theorem 4.2.c is standard. Its unique irreducible quotient is L ( P ( ξ ) , π P ( ξ ) ( ξ, ρ )).By Proposition 3.20.c,e, every standard H -module is of the form (4.3), for some ξ ∈ Ξ + and ρ ∈ Irr( C [ R ξ , (cid:92) ξ ]).When ξ, ξ (cid:48) ∈ Ξ + , ρ ∈ Irr( C [ R ξ , (cid:92) ξ ]) and w ∈ W Ξ with w ( ξ ) ∼ = ξ (cid:48) , we can define ρ (cid:48) ∈ Irr( C [ R w ( ξ ) , (cid:92) w ( ξ ) ]) by ρ (cid:48) (cid:0) π ( w (cid:48) , ξ (cid:48) ) (cid:1) := ρ (cid:0) π ( w, ξ ) − π ( w (cid:48) , ξ (cid:48) ) π ( w, ξ ) (cid:1) . Although π ( w, ξ ) is only defined up to a scalar, the formula for ρ (cid:48) is independent ofthe choice of a normalization.We denote this ρ (cid:48) by w ( ρ ), and we say that ( ξ, ρ ) and ( w ( ξ ) , w ( ρ )) are W Ξ -associate. It is not clear whether this comes from a groupoid action on a set con-taining all ( ξ, ρ ) as above, because W Ξ does not stabilizes the set of positive inductiondata Ξ + and we did not define R-groups for non-positive induction data.Let us summarise some properties of standard H -modules. Corollary 4.4.
Write Ξ + e = { ( ξ, ρ ) : ξ ∈ Ξ + , ρ ∈ Irr( C [ R ξ , (cid:92) ξ ]) } . ( a ) The set of standard H -modules (up to isomorphism) is parametrized by Ξ + e upto W Ξ -association. ( b ) Every standard H -module has a unique irreducible quotient. ( c ) For every irreducible H -representation π there is a unique (up to isomorphism)standard H -module that has π as quotient. ( d ) There are bijections Ξ + e / W Ξ → { standard H -modules } → Irr( H )( ξ, ρ ) (cid:55)→ π ( ξ, ρ ) (cid:55)→ L ( P ( ξ ) , π P ( ξ ) ( ξ, ρ )) . Proof. (a) follows from Theorem 3.23.(b) is already contained in Theorem 4.2.(c) Recall from the remark after Theorem 3.24 that π determines a unique parabol-ically induced representation π ( ξ ), with ξ ∈ Ξ + , that has π as quotient.(d) is a consequence of parts (a), (b) and (c). (cid:3) Corollary 4.4 provides a classification of Irr( H ) in terms of induction data andR-groups. The role of the R-groups is quite subtle, firstly because it can be hard todetermine them, secondly because potentially a non-trivial 2-cocycle can be involvedin the H -endomorphism algebra of a parabolically induced representation.Sometimes it is easier to work with standard modules than with irreducible rep-resentations, for their structure is more predictable. Example 4.5. • For H ( R , q ) with R of type (cid:102) A , almost all standard modulesare irreducible. The only reducible standard H -module is π ( ∅ , triv , q ) =ind H C [ X ] ( C q ), which has triv as irreducible quotient. • For H n ( q ) all the groups R ξ are trivial, so the standard modules are justthe parabolically induced representations π ( ξ ) with ξ ∈ Ξ + . • When H = H ( R ,
1) = C [ X (cid:111) W ] (any R ), there is a standard module π ( ∅ , triv , t, ρ ) for every t ∈ T and ρ ∈ Irr( W t ). In the notation from Para-graph 2.1 it equals π ( t, ρ ∗ ). In view of Theorem 2.2, these H -representationsare irreducible and there are no other standard modules.To the best of our knowledge, the theory of R-groups for graded Hecke algebrashas never been written down explicitly. It can be deduced readily from [DeOp2] andthe algebra isomorphisms from Theorem 3.29. In this setting the c α -function for oneroot α ∈ R becomes ˜ c α ( λ ) = (cid:104) α, λ (cid:105) + k ( α ) (cid:104) α, λ (cid:105) λ ∈ t . For every ˜ ξ = ( P, δ, λ ) ∈ ˜Ξ + we obtain an R-group R ˜ ξ and a 2-cocycle (cid:92) ˜ ξ such thatEnd H ( π ( ˜ ξ )) ∼ = C [ R ˜ ξ , (cid:92) ˜ ξ ] . The standard module associated to ˜ ξ ∈ ˜Ξ + and ρ ∈ Irr( C [ R ˜ ξ , (cid:92) ˜ ξ ]) is(4.4) π ( ˜ ξ, ρ ) = Hom C [ R ˜ ξ ,(cid:92) ˜ ξ ] ( ρ, π ( ˜ ξ )) . Now Theorem 4.2 and Corollary 4.4 apply to H , and they provide bijections (cid:8) ( ˜ ξ, ρ ) : ˜ ξ ∈ ˜Ξ + , ρ ∈ Irr( C [ R ˜ ξ , (cid:92) ˜ ξ ]) (cid:9)(cid:14) W ˜Ξ → { standard H -modules } → Irr( H )( ˜ ξ, ρ ) (cid:55)→ π ( ˜ ξ, ρ ) (cid:55)→ irreducible quotient of π ( ˜ ξ, ρ ) . (4.5)4.2. Residual cosets.
The most significant step towards the classification of discrete series H -representations is the determination of their central characters, which was achievedby Opdam [Opd2]. Consider the following rational function on T :(4.6) c R = (cid:89) α ∈ R c α = (cid:89) α ∈ R (cid:0) θ α − q ( − λ ∗ ( α ) − λ ( α )) / (cid:1)(cid:0) θ α + q ( λ ∗ ( α ) − λ ( α )) / (cid:1) ( θ α − θ α + 1) . Its counterpart for H ( t , W, k ) is˜ c R = (cid:89) α ∈ R ˜ c α = (cid:89) α ∈ R ( α + k ( α )) α − . Definition 4.6.
Let L ⊂ T be a coset of a complex algebraic subtorus of T . Wecall L a residual coset (with respect to R and q ) if the zero order of c R along L isat least the (complex) codimension of L in T .An affine subspace l ⊂ a is called a residual subspace (with respect to R and k )if the zero order of ˜ c R along l is at least the (real) codimension of l in a .A residual point is a residual coset/subspace of dimension zero.For any L or l as above, its zero order is always at most its codimension [Opd2,Corollary A.12]. Hence we may replace ”is at least” by ”equals” in Definition 4.6.This also implies that residual points can only exist if R spans a ∗ . The collection ofresidual cosets/subspaces is stable under W , because R and q/k are so. FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 45
Example 4.7. • T itself is always a residual coset, and a itself is always aresidual subspace. • There are no residual points for H (resp. for H ) if R (cid:54) = ∅ and λ = λ ∗ = 0(resp. k = 0). • Consider a = a ∗ = { x ∈ R n : x + · · · + x n = 0 } , R = A n − = { e i − e j : i (cid:54) = j } , W = S n , k ( α ) = k ∈ R × . There is just one S n -orbit of residual points for H , and it contains( k (1 − n ) / , k (3 − n ) / , . . . , k ( n − / . This point is the O ( t )-character of the Steinberg representation of H , whichby definition is onedimensional and restricts on C [ W ] to the sign represen-tation. • Take a = a ∗ = R , R = B = {± e ± e }∪{± e i } , W = W ( B ) ∼ = D , k ( ± e ± e ) = k , k ( ± e i ) = k . There are at most two W -orbits of residual points for H ( C , W ( B ) , k ), represented by ( k + k , k ) and ( k − k , k ). These pointsare indeed residual if k k ( k + 2 k )( k + k )( k − k )( k − k ) (cid:54) = 0 . The crucial property of residual points is:
Theorem 4.8. [Opd2, Lemma 3.31]
Let δ ∈ Irr( H ) be discrete series. Then all its C [ X ] -weights are residual points for ( R, q ) . Conversely, if t ∈ T is a residual point for ( R, q ) , then there exists a discreteseries H -representation with central character W t . Example 4.9.
Consider the root datum R of P GL n ( C ), with X = Z n / Z (1 , , . . . , , Y = { y ∈ Z n : y + · · · + y n = 0 } ,R = R ∨ = A n − and W = S n . Notice that T W ∼ = Z /n Z , generated by ζ n : x (cid:55)→ exp(2 πi ( x + · · · + x n )) . For q (cid:54) = 1, H ( R , q ) admits a unique S n × T W -orbit of residual points, one such pointbeing t q = (cid:0) q (1 − n ) / , q (3 − n ) / , . . . , q ( n − / (cid:1) . For q >
1, this is the unique C [ X ]-weight of the Steinberg representation of H ( R , q )–which is defined just like St for H n ( q ) in (2.11) and (2.13). Similarly ζ n t q is the C [ X ]-weight of the discrete series representation St ⊗ ζ n .There is a general method to construct residual cosets from residual points forsubquotient algebras. Namely, let P ⊂ ∆ and let r ∈ T P be a residual point for H P .Then T P r is a residual coset for H [Opd2, Proposition A.4]. Up to the action of W ,every residual coset is of this form.From that, Theorem 4.8 and Lemma 3.3 we deduce: for any induction datum( P, δ, t ) ∈ Ξ, every weight of π ( P, δ, t ) lies in a residual coset of the same dimensionas T P .Now we relate residual cosets to residual subspaces for graded Hecke algebras.By definition every residual coset for H can be written as L = u exp( λ ) T L , where u ∈ T u n, λ ∈ a and T L ⊂ T is a complex algebraic subtorus. Proposition 4.10. [Opd2, Theorem A.7] ( a ) With the above notations, λ + log | T L | ⊂ a is a residual subspace for H ( t , W ( R u ) , k u ) . ( b ) Every residual subspace for H ( t , W ( R u ) , k u ) arises in this way. ( c ) When T L + T P and u exp( λ ) is a residual point for H P , λ ∈ a P is a residual pointfor H ( t P , W ( R P,u ) , k u ) . In this case R P,u has the same rank as R P , namely | P | . Example 4.11.
Take X = Y = Z , R = B , R ∨ = C = {± e ± e } ∪ {± e , ± e } and ∆ = { e − e , e } . We write q = q λ ( ± e ± e ) , q = q λ ( ± e i ) , q = q λ ∗ ( ± e i ) . The points u ∈ T un with R u of rank 2 are (1 , , ( − , − , (1 , −
1) and ( − , W -associate. There are at most 5 W -orbits of residual points, summarisedin the following table: u (1 ,
1) ( − , −
1) (1 , − R u B B A × A k u log( q ) , k = log( q q )2 log( q ) , k = log( q q − )2 log( q q )2 , log( q q − )2residual ( q − e − k , e − k ) ( − q − e − k , − e − k ) (( q q ) − / , points ( q − e k , e − k ) ( − q − e k , − e − k ) − q − / q / )Here we give k u in terms of its values on a basis of R u . For generic parameters q , q , q , all the five points of T in the above table are indeed residual, and eachof them represents the central character of a unique discrete series representation[Sol4, § H can be translated to residual subspaces for graded Hecke algebras.Combining that with Lemma 3.32, we find that every O ( t )-weight of a discrete se-ries H -representation is a residual point in a . Example 4.12.
We continue Example 4.11, but now for the graded Hecke algebra H built from a = a ∗ = R , t = t ∗ = C , R = B , W = W ( B ), ∆ = { α = e − e , β = e } , k ( ± e ± e ) = k > , k ( ± e i ) = k >
0. The residual point( − k − k , − k ) is the O ( t )-character of the Steinberg representation of H , whichis discrete series and restricts to the sign character of C [ W ]. When k < k , theresidual point ( k − k , − k ) is the O ( t )-character of a onedimensional discrete series H -representation δ . Its restriction to C [ W ] is given by δ ( s e ) = − , δ ( s e − e ) = 1.With Theorem 3.29 and Corollary 4.4 we can complete the classification of Irr( H ).To this end we note that H { α } = H ( C α, (cid:104) s α (cid:105) , k ) and H { β } = H ( C β, (cid:104) s β (cid:105) , k ) . These algebras have a unique discrete series representation, namely St. Further t { α } = C ( e + e ) , a { α } + = R ≥ ( e + e ) , t { β } = C e , a { β } + = R ≥ e and a + = R ≥ e + R ≥ ( e + e ). All the R-groups R ˜ ξ for H are trivial, so (4.5)provides a bijection from ˜Ξ + / W ˜Ξ to Irr( H ), where˜Ξ + = (cid:8) ( ∅ , triv , λ ) : λ ∈ i a + a + (cid:9) ∪ (cid:8) ( { α } , St , λ ) : λ ∈ i a { α } + a { α } + (cid:9) ∪ (cid:8) ( { β } , St , β ) : λ ∈ i a { β } + a { β } + (cid:9) ∪ { St , δ } . FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 47
The groupoid W ˜Ξ consists of the groups W ˜Ξ , ∅∅ = W, W ˜Ξ , { α }{ α } = (cid:104) s e + e (cid:105) , W ˜Ξ , { β }{ β } = (cid:104) s e (cid:105) , W ˜Ξ , ∆∆ = { id } . The action of W ˜Ξ on ˜Ξ makes some of the ( P, δ, λ ) ∈ ˜Ξ + with (cid:60) ( λ ) ∈ ∂ ( a P + ) W ˜Ξ -associate, for instance ( ∅ , triv , (1 , i )) and ( ∅ , triv , (1 , − i )).In general it is easy to classify all points u ∈ T un for which R u has full rank in R , in terms of the affine Dynkin diagram of R [Opd2, Lemma A.8]. Recall that therelation between the representations of H ( t , W ( R u ) , k u ) and of H ( t , W ( R u ) , k u ) (cid:111) Γ u is well-understood from Clifford theory. Thus the classification of discrete series H -representation boils down to two tasks: • classify all residual points for ( t , W, k ), where k is any real-valued parameterfunction, • for a given residual point λ ∈ a , classify the discrete series H -representationswith central character W λ .In view of the isomorphism (1.25), it suffices to this when R is irreducible. Theresidual points for H ( t , W, k ) with such R and k have been classified completely in[HeOp, § f ( k ) in the parameters k ( α ) for α ∈ R . For a given such f , f ( k ) is residual with respect to R and k for almost all k : R → R . We say that a parameter function k is generic if all potentially residualpoints f ( k ), for the H ( t , W ( R P ) , k ) with P ⊂ ∆, are really residual for this k , andare all different. Theorem 4.13. [OpSo2, Theorems 3.4 and 7.1]
Let R ⊂ a ∗ be an irreducible root system that spans a ∗ . ( a ) Let k : R → R be a generic parameter function. The central character mapgives a bijection from the set of irreducible discrete series representations of H ( t , W ( R ) , k ) to the set of W -orbit of residual points for R and k (except when R ∼ = F , then one fiber of this map has two elements). ( b ) For a non-generic parameter function k (cid:48) : R → R and a residual point λ ∈ a ,consider the collection of generic residual points { f i ( k ) } i that specialize to ξ at k (cid:48) = k . For k close to k (cid:48) in the space of all parameter functions R → R , thereis a natural bijection between: • the set of irreducible discrete series representations of H ( t , W ( R ) , k ) withcentral character in { W f i ( k ) } i , • the set of irreducible discrete series representation of H ( t , W ( R ) , k (cid:48) ) withcentral character W λ .More explicitly, every H ( t , W ( R ) , k ) -representation of the indicated kind is ofpart of a unique continuous family of such representations, one representationfor each k in some neighborhood of k . The above bijection matches the membersof such a continuous family at k and at k (cid:48) . From Theorem 4.13 one obtains a complete classification of discrete series rep-resentations of affine Hecke algebras with positive parameters. However, we haveto point out that this does not yet achieve an actual classification of all irreduciblerepresentations. The problem is that it can remain difficult to effectively computethe R-groups R P,δ,t and their 2-cocycles (cid:92)
P,δ,t from Paragraph 4.1.
We conclude this section with a discussion of the residual points for H in the mostintricate case, for root systems of type B n . We take a = a ∗ = R n , R = B n = {± e i : i = , . . . , n } ∪ {± e i ± e j : i (cid:54) = j } and we write k ( ± e i ± e j ) = k , k ( ± e i ) = k . Forevery partition (cid:126)n = ( n , n , . . . , n d ) of n we construct a point λ ( (cid:126)n, k ) ∈ a , or rathera S n -orbit in a , in the following way. Draw the Young diagram, with first columnof n boxes, second column of n boxes and so on. Label the boxes from b to b n in some way (to does not matter how, for another labelling will produce a point inthe same S n -orbit in a ). We define the height of a box b in column i and row j tobe h ( b ) = j − i and we write(4.7) λ ( (cid:126)n, k ) = ( h ( b ) k + k , h ( b ) k + k , . . . , h ( b n ) k + k ) . For example, when n = 2 we have λ ((2) , k ) = ( k + k , k ) , λ ((1 , , k ) = ( − k + k , k )Every residual point for H ( C n , B n , k ) is W ( B n )-associate to a λ ( (cid:126)n, k ) [HeOp, Propo-sitions 4.3 and 4.5]. For most parameters k i indeed all these points of a are residual,but not for all parameters. An extreme case is k = k = 0, then there are noresidual points.A parameter function k : B n → R is generic if(4.8) k k (cid:89) n − j =1 ( jk + 2 k )( jk − k ) (cid:54) = 0 . When k is generic all the λ ( (cid:126)n, k ) are residual, and they belong to different W ( B n )-orbits. On the other hand, when k is not generic, some of the λ ( (cid:126)n, k ) are not residual,and some of them may belong to the same W ( B n )-orbit.5. Geometric methods
We survey some of results on affine Hecke algebras obtained with methods fromcomplex algebraic geometry. In many cases, these provide a complete classificationof standard modules and of irreducible representations.5.1.
Equivariant K-theory.
In this paragraph we discuss equal label affine Hecke algebras, that is, with a singleparameter q . Recall from Paragraph 1.2 that these algebras are especially importantbecause they classify representations of reductive p -adic groups with vectors fixedby an Iwahori subgroup. It was realized independently by Kazhdan–Lusztig andby Ginzburg that such affine Hecke algebras can be realized as the equivariant K-theory of a suitable complex algebraic variety. Then its representations can beanalysed in algebro-geometric terms, and that leads to a beautiful construction andparametrization of all irreducible representations.Let G be a connected complex reductive group with a maximal torus T , and let R ( G, T ) be the associated root datum. We define H ( G, T ) to be like H ( R ( G, T ) , q ),but with q replaced by an invertible formal variable z . As vector spaces H ( G, T ) = C [ X ∗ ( T )] ⊗ C C [ W ] ⊗ C C [ z , z − ] , where W = W ( G, T ). The upcoming constructions work best when the derivedgroup of G is simply connected, so we assume that in most of this paragraph. Amain role is played by the Steinberg variety of G : Z := { ( B, u, B (cid:48) ) :
B, B (cid:48)
Borel subgroups of
G, u ∈ B ∩ B (cid:48) unipotent } . FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 49
The group G × C × acts on Z by( g, λ )( B, u, B (cid:48) ) = ( gBg − , gu λ − g − , gB (cid:48) g − ) . Note that u λ − is defined because u is unipotent. This C × -action might appearad hoc, but it is indispensable to obtain Hecke algebras. Without it, we could atbest build the G -equivariant K-group K G ( Z ), which turns out be isomorphic to C [ X ] (cid:111) W [ChGi, Theorem 7.2]. According to [KaLu2, Theorem 3.5] and [ChGi,Theorem 7.2.5], there is a natural isomorphism(5.1) K G × C × ( Z ) ∼ = H ( G, T ) . The z ’s in H ( G, T ) are due to the C × -action on Z . The ring of regular class functionson G is R ( G ) = O ( G ) G ∼ = O ( T /W ) . When we regard z as the identity representation of C × , we can write the rings ofregular class functions on C × and on G × C × as R ( C × ) = C [ z , z − ] , R ( G × C × ) = O ( G ) G ⊗ C C [ z , z − ] . By construction [ChGi, § R ( G × C × ) acts naturally on K G × C × ( Z ). On the otherhand, a variation on (1.19) shows that R ( G × C × ) is also naturally isomorphic to thecentre of H ( G, T ). With these identifications the isomorphism (5.1) is R ( G × C × )-linear. For any q ∈ C × we can specialize (5.1) to an isomorphism(5.2) K G × C × ( Z ) ⊗ C [ z , z − ] C q ∼ = H ( R ( G, T ) , q ) . Further, let t ∈ G be a semisimple element and denote the associated onedimensionalrepresentation of R ( G × C × ) by C t, q . Let Z t,q be the subvariety of Z fixed by( t, q ) ∈ G × C × . By [ChGi, p. 414] there is an isomorphism(5.3) K G × C × ( Z ) ⊗ R ( G × C × C t, q ∼ = K ( Z t, q ) ⊗ Z C ∼ = H ∗ ( Z t, q , C ) . The construction of K G × C × ( Z )-modules is performed most naturally with Borel–Moore homology (that is equivalent to the constructions with equivariant K-theoryin [KaLu2]).Let u ∈ G be a unipotent element such that tut − = u q . Let B be the variety ofBorel subgroups of G and let B t,u be the subvariety of Borel subgroups that contain t and u . The convolution product in Borel–Moore homology [ChGi, Corollary 2.7.42]provides an action of H ∗ ( Z t, q , C ) on H ∗ ( B t,u , C ). This and (5.3) make H ∗ ( B t,u , C )into a K G × C × ( Z )-module, usually reducible.By [ChGi, Lemma 8.1.8] these constructions commute with the G -action, in thesense that H ∗ (Ad g ) ∗ : H ∗ ( B t,u , C ) → H ∗ ( B gtg − ,gug − , C )intertwines the K G × C × ( Z )-actions. In particular Z G ( t, u ) acts on H ∗ ( B t,u , C ) by K G × C × ( Z )-intertwiners. The neutral component of Z G ( t, u ) acts trivially, so wemay regard it as an action of the component group π ( Z G ( t, u )). That can be usedto decompose the module H ∗ ( B t,u , C ). Let ρ be an irreducible representation of π ( Z G ( t, u )) which occurs in H ∗ ( B t,u , C ). Then K t,u,ρ := Hom π ( Z G ( t,u )) (cid:0) ρ, H ∗ ( B t,u , C ) (cid:1) is a nonzero K G × C × ( Z )-module, called standard in [KaLu2, 5.12] and [ChGi, Defi-nition 8.1.9]. Since the action factors via (5.3), K t,u,ρ can be regarded as a H ( R ( G, T ) , q )-representation with central character W t . We call data ( t, u, ρ ) withthe above properties a Kazhdan–Lusztig triple for ( G, q ). Theorem 5.1. [KaLu2, Theorem 7.12] and [ChGi, Theorem 8.1.16]
Let q ∈ C × be of infinite order. ( a ) For every Kazhdan–Lusztig triple ( t, u, ρ ) , the H ( R ( G, T ) , q ) -module K t,u,ρ hasa unique irreducible quotient L t,u,ρ . ( b ) Every irreducible H ( R ( G, T ) , q ) -module is of the form L t,u,ρ , for a suitableKazhdan–Lusztig triple. ( c ) Let ( t (cid:48) , u (cid:48) , ρ (cid:48) ) be another Kazhdan–Lusztig triple. Then L t,u,ρ ∼ = L t (cid:48) ,u (cid:48) ,ρ (cid:48) if andonly if there exists a g ∈ G such that t (cid:48) = gtg − , u (cid:48) = gug − and ρ (cid:48) = ρ ◦ Ad( g − ) . This major result comes with a lot of extras. Firstly, suppose that L is a standardLevi subgroup of G and that it contains { t, u } . Then K L × C × ( Z L ) ⊗ C [ z , z − ] C q ∼ = H ( R ( L, T ) , q )embeds naturally in H ( R ( G, T ) , q ) and, by [KaLu2, Theorem 6.2]: H ∗ ( B t,u , C ) ∼ = ind H ( R ( G,T ) , q ) H ( R ( L,T ) , q ) H ∗ ( B t,uL , C ) . For the second extra we suppose that q ∈ R > . By [KaLu2, Theorem 8.3] and[ABPS1, Proposition 9.3] the H ( R ( G, T ) , q )-module L t,u,ρ is essentially discrete se-ries if and only if { t, u } is not contained in any Levi subgroup of any proper parabolicsubgroup of G .With these two extras at hand, we can compare K t,u,ρ with the more analyticapproach from Sections 3 and 4. Let L be a Levi subgroup of G which contains { t, u } and is minimal for that property. Upon conjugating everything by an elementof G , we may assume that L is standard, that t ∈ T and that log | t | is as positive aspossible in its W -orbit. Let ρ L ∈ Irr (cid:0) π ( Z L ( t, u )) (cid:1) be an irreducible constituent of ρ | π ( Z L ( t,u )) . Then(5.4) Hom π ( Z L ( t,u )) (cid:0) ρ L , H ∗ ( B t,uL , C ) (cid:1) is an irreducible essentially discrete series H ( R ( L, T ) , q )-representation. By Theo-rem 3.18 it is of the form δ ◦ ψ t (cid:48) , where δ is discrete series. Then ξ = (∆ L , δ, t (cid:48) ) ∈ Ξ + by the assumption on t . The induction of (5.4) to H ( R ( G, T ) , q ) contains K t,u,ρ asa direct summand. In view of Theorem 4.2.b, this summand must be picked out byan irreducible representation ρ (cid:48) of C [ R ξ , (cid:92) ξ ]. We conclude that K t,u,ρ ∼ = π ( ξ, ρ (cid:48) ), astandard module in the sense of Definition 3.12.This argument also works in the opposite direction, and then it shows that thestandard modules from Definition 3.12 are precisely the standard modules from[KaLu2] and [ChGi].Recall that in Theorem 5.1 the complex reductive group G has simply connectedderived group G der . Without that assumption on G , K G × C × behaves less well.Nevertheless, the parametrization of irreducible H ( R ( G, T ) , q )-representations ob-tained in Theorem 5.1 is valid for any complex reductive group G . This was shownby Reeder [Ree2, Theorem 3.5.4], via reduction to the case with simply connected G der .When q is a root of unity, Theorem 5.1 can definitely fail, in particular when q isa zero of the Poincar´e polynomial of W . On the other hand, if q is not a zero of the FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 51 polynomials (1.9) for any finite reflection subgroup of X (cid:111) W , then it seems likelythat large parts of the K-theoretic approach are still valid.A very special case arises when q = 1. Then Theorem 5.1 holds, in a slightlydifferent, simpler way [Kat2]. In this case the action of H ( R ( G, T ) ,
1) = C [ X ∗ ( T )] (cid:111) W on K t,u,ρ preserves the homological degree, so Hom π ( Z G ( t,u )) (cid:0) ρ, H d ( B t,u , C ) (cid:1) is asubrepresentation for any d ∈ Z ≥ . Then K t,u,ρ may obviously have many irreduciblequotients, so the previous definition of L t,u,ρ cannot be used anymore. Instead wedefine(5.5) L t,u,ρ = Hom π ( Z G ( t,u )) (cid:0) ρ, H dim R ( B t,u ) ( B t,u , C ) (cid:1) when q = 1 , that is, we only use the homology in the largest possible degree. It can be shownthat (5.5) is the canonical irreducible quotient of K t,u,ρ [ABPS1, § q = 1 [Kat2, Theorem 4.1]. Notice that herethe triples ( t, u, ρ ) satisfy tu = ut , so u is a unipotent element of Z G ( t ). This classi-fication of Irr( X (cid:111) W ) can be regarded as a Springer correspondence for affine Weylgroups. The proof is much shorter than that of Theorem 5.1, it mainly relies on theSpringer correspondence for finite Weyl groups.As already observed in [KaLu2, § G -conjugation (as inTheorem 5.1.c). One alternative shows the connection between different q ’s verynicely.Let ( t , u, ρ ) be a Kazhdan–Lusztig triple for ( G, φ : SL ( C ) → Z G ( t ) with φ (( )) = u . By the Jacobson–Morozovtheorem, such a φ exists and is unique up to conjugation by Z G ( t , u ). Assumethat we have a preferred square root of q . We put t q = t φ ( (cid:16) q / q − / (cid:17) ), so that t q ut − q = u q . Then B t ,u and B t q ,u are homotopy equivalent and ρ gives rise toa unique ρ q [ABPS1, Lemma 6.1]. This provides a bijection between G -conjugacyclasses of Kazhdan–Lusztig tripes for ( G,
1) and for ( G, q ) [ABPS1, Lemma 7.1]. Incombination with Theorem 5.1 we obtain: Corollary 5.2.
Let q ∈ C × be either 1 or not a root of unity. There exists acanonical bijection { Kazhdan–Lusztig triples for ( G, } /G ←→ Irr (cid:0) H ( R ( G, T ) , q ) (cid:1) ( t , u, ρ ) (cid:55)→ L t q ,u,ρ q One advantage of this parametrization is that the pair ( t , u ) is the Jordan decom-position of an arbitrary element of G , so that ( t , u ) up to G -conjugacy parametrizesthe conjugacy classes of G . Example 5.3.
Let R be of type GL n and consider H n ( q ). As Z GL n ( C ) ( t , u ) = Z GL n ( C ) ( t u ) is always connected, ρ is necessarily trivial and may be ignored. Corol-lary 5.2 recovers the parametrization of Irr( H n ( q )) summarised in Theorem 2.6.In addition to the already discussed properties of these bijections, we mentionthat temperedness can be detected easily in Corollary 5.2. Namely, by [ABPS1,Proposition 9.3], for q ∈ R ≥ :(5.6) L t q ,u,ρ q is tempered ⇐⇒ t lies in a compact subgroup of G. Equivariant homology.
The material in the previous paragraph learns us a lot about equal label affineHecke algebras, but very little about the cases with several parameters q s . It ap-pears that equivariant K-theory is not flexible enough to incorporate more than oneindependent q -variable.Faced with this problem, Lusztig discovered that Hecke algebras with multipleparameters can still be studied geometrically, if we accept two substantial modifica-tions: • replace affine Hecke algebras by their graded versions, • replace equivariant K-theory by equivariant homology.Not all combinations of q -parameters can obtained in this way, but still a con-siderable number of them. Our treatment of this method is based on the papers[Lus1, Lus2, Lus4, Lus6, AMS2].Let G be a connected complex reductive group and let P be a parabolic subgroupof G with Levi factor L and unipotent radical U . We denote the Lie algebras ofthese groups by g , p , l and u . Let v ∈ l be nilpotent and let C Lv be its adjointorbit. Let L be an irreducible L -equivariant cuspidal local system on C Lv . We refrainfrom explaining these notions here, instead we refer to [Lus1], where cuspidal localsystems are introduced and classified.To the above data we will associate a graded Hecke algebra. We take T = Z ( L ) ◦ ,a (not necessarily maximal) torus in G . According to [Lus2, Proposition 2.5], thecuspidality of L implies that: • the set of weights of T acting on g is a (possibly nonreduced) root system R ( G, T ) in X ∗ ( T ), • the Weyl group of R ( G, T ) is W L := N G ( L ) /L = N G ( T ) /Z G ( T ).The parabolic subgroup P determines a basis ∆ L of R ( G, T ). Let t = X ∗ ( T ) ⊗ Z C be the Lie algebra of T , so that R ( G, T ) ⊂ t ∗ = X ∗ ( T ) ⊗ Z C . The action of W L on T induces actions on t and t ∗ , which stabilize R ( G, T ). Thedefinition of the parameter function k : R ( G, T ) → Z involves the nilpotent element v . Let g α ⊂ g be the root space and let s α ∈ W L be the reflection associated to α ∈ R ( G, T ). Since v ∈ l commutes with t , ad( v ) stabilizes each g α . For α ∈ ∆ L one defines k ( α ) ∈ Z ≥ byad( v ) k ( α ) − : g α ⊕ g α → g α ⊕ g α is nonzero,ad( v ) k ( α ) − : g α ⊕ g α → g α ⊕ g α is zero.Then k ( α ) = k ( β ) whenever α, β ∈ ∆ L are W L -associate. Now we can define H ( G, L, L , r ) = H ( t , W L , k, r ) . Suppose that G is an almost direct product of connected normal subgroups G and G . Then L, C Lv and L decompose accordingly and H ( G, L, L , r ) = H ( G , L , L , r ) ⊗ C [ r ] H ( G , L , L , r ) . If G is a torus, then necessarily L = T = G and v = 0. In that case L is trivial and H ( T, T, L , r ) is just O ( t ) ⊗ C C [ r ]. Hence the study of H ( G, L, L ) can be reduced to FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 53 simple G . Then it becomes feasible to classify the data, and indeed this has beendone in [Lus2, 2.13]. We tabulate the possibilities for g , l , R ( G, T ) and k :(5.7) g l R ( G, T ) k simple Cartan irreducible k ( α ) = 2 sl ( d +1) p s l d +1 p ⊕ C d A d k ( α ) = 2 p sp n +2 d , n = p ( p + 1) sp n ⊕ C d BC d k ( α ) = 2 , k ( β ) = 2 p + 1 so n +2 d , n = p so n ⊕ C d B d k ( α ) = 2 , k ( β ) = 2 p so n +4 d , n = p (2 p − sl d ⊕ C d BC d k ( α ) = 4 , k ( β ) = 4 p − so n +4 d , n = p (2 p + 1) sl d ⊕ C d BC d k ( α ) = 4 , k ( β ) = 4 p + 1 E sl ⊕ C G k ( α ) = 2 , k ( β ) = 6 E sl ⊕ C F k ( α ) = 2 , k ( β ) = 4In this table d, p ∈ Z > are arbitrary, α ∈ R ( G, T ) is a long root and β ∈ R ( G, T ) isa short root. (For R ( G, T ) of type BC d we mean that α = ± e i ± e j and β = ± e i .)Recall from (1.22) that we can simultaneously rescale all the k ( α ) without chang-ing the algebra (up to isomorphism). When R ( G, T ) has roots of different lengths,we can also adjust the parameters for roots of one length in a specific way:
Example 5.4.
Let R be of type B n , F or G . Assume that R spans a ∗ . Anyparameter function k for R has two independent values k = k ( α ) and k = k ( β ),which can be chosen arbitrarily. We write k = ( k , k ) and we consider the gradedHecke algebra H ( t , W ( R ) , k , k , r ).Take (cid:15) = 2 for B n or F , and (cid:15) = 3 for G . The set { wα : w ∈ W } ∪ { (cid:15)wβ : w ∈ W } is a root system in a ∗ , of type (respectively) C n , F or G . Notice that now α is shortand (cid:15)β is long. The identity map on the vector space underlying H ( t , W ( R ) , k , k , r )provides an algebra isomorphism H ( C n , W ( B n ) , k , k , r ) → H ( C n , W ( C n ) , k , k , r )(5.8) H ( C , W ( F ) , k , k , r ) → H ( C , W ( F ) , k , k , r )(5.9) H ( C , W ( G ) , k , k , r ) → H ( C , W ( G ) , k , k , r )(5.10)In particular any graded Hecke algebra of type C n is also a graded Hecke algebra oftype B n (but with different parameters).We will call a parameter function obtained from the above table by a compositionof the isomorphisms (1.22) and those from Example 5.4 geometric. Thus we have alarge supply of geometric parameter functions for type B/C root systems.Next we describe the geometric realization of H ( G, L, L , r ). We need the varieties˙ g = { ( x, gP ) ∈ g × G/P : Ad( g − ) x ∈ C Lv + t + u } , ¨ g N = { ( x, gP, g (cid:48) P ) ∈ g × ( G/P ) : ( x, gP ) ∈ ˙ g , ( x, g (cid:48) P ) ∈ ˙ g , x nilpotent } . The first is a variation on the variety B of Borel subgroups of G , while the secondhas a flavour of the Steinberg variety of G . The group G × C × acts on ˙ g by( g , λ )( x, gP ) = ( λ − Ad( g ) x, g gP ) , and similarly on ¨ g N . The L × C × -equivariant local system L on C Lv yields a G × C × -equivariant local system ˙ L on ˙ g . The two projections ¨ g N → ˙ g give rise toan equivariant local system ¨ L = ˙ L (cid:2) ˙ L ∗ on ¨ g N . Equivariant (co)homology with coefficients in a local system is defined in [Lus2, § § H G × C × ∗ ( ¨ g N , ¨ L ) into a gradedalgebra, see the proof of [Lus4, Theorem 8.11]. Theorem 5.5. [Lus2, Corollary 6.4] and [Lus4, Theorem 8.11]
There exists a canonical isomorphism of graded algebras H ( G, L, L , r ) −→ H G × C × ∗ ( ¨ g N , ¨ L ) . With equivariant homology one can construct many modules for H G × C × ∗ ( ¨ g N , ¨ L ).Let y ∈ g be nilpotent and define P y = { gP ∈ G/P : Ad( g − ) y ∈ C Lv + u } . This is the appropriate analogue of the variety B u of Borel subgroups containing u .The group M ( y ) := { ( g , λ ) ∈ G × C × : Ad( g ) y = λ y } acts on P y by ( g , λ ) gP = g gP . The inclusion { y } × P y → ˙ g is M ( y )-equivariant,which allows us to restrict ˙ L to an equivariant local system on P y . With con-structions in equivariant (co)homology [AMS2, § H ( G, L, L , r ) on H M ( y ) ◦ ∗ ( P y , ˙ L ). It commutes with the natural actions of π ( M ( y ))and of H ∗ M ( y ) ◦ ( { y } ) on H M ( y ) ◦ ∗ ( P y , ˙ L ) which enables us to decompose it as H ( G, L, L , r )-module. It is known that H M ( y ) ◦ ∗ ( P y , ˙ L ) is projective over H ∗ M ( y ) ◦ ( { y } ).One can naturally identify H ∗ M ( y ) ◦ ( { y } ) = O (Lie( M ( y ) ◦ )) M ( y ) ◦ , Lie( M ( y ) ◦ ) = { ( σ, r ) ∈ g ⊕ C : [ σ, y ] = 2 ry } . In particular the characters of H ∗ M ( y ) ◦ ( { y } ) are parametrized by semisimple adjointorbits in Lie( M ( y ) ◦ ). For a semisimple element ( σ, r ) ∈ Lie( M ( y ) ◦ ) we have the H ( G, L, L , r )-module(5.11) E y,σ,r = C σ,r ⊗ H ∗ M ( y ) ◦ ( { y } ) H M ( y ) ◦ ∗ ( P y , ˙ L ) . By the projectivity of H M ( y ) ◦ ∗ ( P y , ˙ L ), the restriction of E y,σ,r to C [ W L ] does notdepend on ( σ, r ). As usual, the isomorphism class of E y,σ,r depends only on ( y, σ, r )up to the adjoint action of G (which fixes r ). From the action of π ( M ( y )) on H M ( y ) ◦ ∗ ( P y , ˙ L ), only the operators that stabilize the adjoint orbit [ σ ] of σ inLie( M ( y ) ◦ ) act on E y,σ,r . Hence irreducible representations ρ of π ( M ) [ σ ] can be usedto decompose the H ( G, L, L , r )-module E y,σ,r further. We define the H ( G, L, L , r )-representation E y,σ,r,ρ = Hom π ( M ( y )) [ σ ] ( ρ, E y,σ,r ) . We call this a standard module if it is nonzero.Let us improve the bookkeeping for the parameters ( y, σ, r, ρ ) just obtained. WithJacobson–Morozov we pick an algebraic homomorphism γ y : SL ( C ) → G such thatd γ y (( )) = y . Notice that the semisimple element σ = σ + d γ y ( (cid:0) − r r (cid:1) )commutes with y . It is not difficult to see that π ( M ( y )) [ σ ] is naturally isomorphicto π ( Z G ( σ , y )). By [AMS2, Lemma 3.6] these constructions provide a bijection FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 55 between G -association classes of data ( y, σ, ρ ) as above (for a fixed r ∈ C ) and G -association classes of triples ( y, σ , ρ ), where(5.12) y ∈ g nilpotent, σ ∈ g semisimple, [ σ , y ] = 0 , ρ ∈ Irr (cid:0) π ( Z G ( σ , y )) (cid:1) . In the previous paragraph we encountered a clear condition on the representation ρ of the component group: it should appear in the homology of a particular variety,otherwise the associated module would be 0.In the current setting the condition on ρ is more subtle, because P y can be emptyand a local system L is involved. To formulate it we need the cuspidal supportmap Ψ G from [Lus1, 6.4]. It associates a cuspidal support ( L (cid:48) , C L (cid:48) v (cid:48) , L (cid:48) ) to every pair( x, ρ (cid:48) ) with x nilpotent and ρ (cid:48) ∈ Irr (cid:0) π ( Z G ( x )) (cid:1) . Giving such ( x, ρ ) is equivalentto giving a G -equivariant cuspidal local system on a nilpotent orbit in g (which isalso an equivariant perverse sheaf). The cuspidal support map can be expressedwith a version of parabolic induction for equivariant perserve sheaves [Aub, § E y,σ,r,ρ (cid:54) = 0 ⇐⇒ Ψ Z G ( σ ) ( y, ρ ) is G -associate to ( L, C Lv , L ) . When L = T is a maximal torus of G and v = 0, we have P y = B exp y and L istrivial. Then the condition (5.13) reduces to: ρ appears in H ∗ ( B exp yZ G ( σ ) , C ). That isequivalent to the condition on ρ in the Kazhdan–Lusztig triple (exp( σ ) , exp( y ) , ρ )for ( G, Theorem 5.6. [Lus6, Theorem 1.15] and [AMS2, Theorem 3.11]
Let ( y, σ , ρ ) be as in (5.12) , such that Ψ Z G ( σ ) ( y, ρ ) is G -associate to ( L, C Lv , L ) .For r ∈ C we write σ r = σ + d γ y ( (cid:0) r − r (cid:1) ) , where γ y : SL ( C ) → Z G ( σ ) withd γ y (( )) = y . ( a ) For r (cid:54) = 0 , E y,σ r ,r,ρ has a unique irreducible quotient, which we call M y,σ r ,r,ρ . ( b ) For r = 0 , E y,σ , ,ρ has a canonical irreducible quotient M y,σ , ,ρ (the directsummand in one specific homological degree). ( c ) Parts (a) and (b) set up a bijection between
Irr (cid:0) H ( G, L, L , r ) / ( r − r ) (cid:1) and the G -association classes of triples ( y, σ , ρ ) as above. ( d ) Every irreducible constituent of E y,σ r ,r,ρ different from M y,σ r ,r,ρ is isomorphicto M y (cid:48) ,σ (cid:48) ,r,ρ (cid:48) , for data ( y (cid:48) , σ (cid:48) , ρ (cid:48) ) as above that satisfy dim C Gy < dim C Gy (cid:48) . Lusztig investigated when the modules E y,σ,r,ρ are tempered or discrete series[Lus6]. Unfortunately his notions differ from ours, and as a consequence the resultingproperties are opposite to what we want. To reconcile it, we use the Iwahori–Matsumoto involution of H ( G, L, L , r ). It is the algebra automorphismIM : H ( G, L, L , r ) → H ( G, L, L , r )IM( N w ) = sign( w ) N w , IM( r ) = r , IM( ξ ) = − ξ w ∈ W L , ξ ∈ t ∗ . Clearly composition with IM has the effect x (cid:55)→ − x on O ( t )-weights of H ( G, L, L , r )-representations, and similarly for central characters. Let y, σ , ρ and γ y be as inTheorem 5.6 and (5.12). We define(5.14) (cid:101) E y,σ ,r,ρ = IM ∗ E y, d γ y (cid:16) r − r (cid:17) − σ ,r,ρ (cid:102) M y,σ ,r,ρ = IM ∗ M y, d γ y (cid:16) r − r (cid:17) − σ ,r,ρ The modules (5.14) enjoy the same properties as their ancestors without tildes inTheorem 5.6. By [Lus4, Theorem 8.13] and [AMS2, Theorem 3.29] all these fourmodules admit the same central character, namely(5.15) (Ad( G )( σ r ) ∩ t , r ) = (Ad( G )( σ − d γ y (cid:0) r − r (cid:1) ) ∩ t , r ) . Theorem 5.7.
Let ( y, σ , ρ ) be as in Theorem 5.6. ( a ) When (cid:60) ( r ) ≥ , the following are equivalent: • (cid:101) E y,σ ,r,ρ is tempered, • (cid:102) M y,σ ,r,ρ is tempered, • Ad( G ) σ intersects i a = i R ⊗ Z X ∗ ( T ) . ( b ) When (cid:60) ( r ) > , the following are equivalent: • (cid:102) M y,σ ,r,ρ is essentially discrete series, • y is distinguished nilpotent in g , that is, not contained in any proper Levisubalgebra of g .Moreover in this case σ ∈ Z ( g ) and (cid:101) E y,σ ,r,ρ = (cid:102) M y,σ ,r,ρ . ( c ) When r ∈ R , the central character of (cid:101) E y,σ ,r,ρ lies in a /W if and only if σ ∈ Ad( G ) a .Proof. (a) and (b) See [AMS2, (84) and (85)].(c) Upon conjugating the parameters by a suitable element of G , we may assumethat σ , d γ y (cid:0) − (cid:1) ∈ t [AMS2, Proposition 3.5.c]. Then d γ y (cid:0) r − r (cid:1) represents thecentral character of the module (cid:101) E y, ,r = IM ∗ E y, d γ y (cid:16) r − r (cid:17) ,r for H (cid:0) Z G ( σ ) , L, L , r (cid:1) / ( r − r ) ∼ = H (cid:0) Z G ( σ ) der , L ∩ Z G ( σ ) der , L , r (cid:1) / ( r − r ) ⊗ C O ( Z g ( σ )) . Part (b) tells us that the restriction of (cid:101) E y, ,r to H ( Z G ( σ ) der , L ∩ Z G ( σ ) der , L , r ) is adirect sum of discrete series representations. By [Slo2, Lemma 2.13] the central char-acters of these representations lie in Z g ( σ ) der ∩ a /W L ∩ Z G ( σ ) der . Hence d γ y (cid:0) r − r (cid:1) ∈ a ,which implies that σ ∈ a ⇐⇒ σ − d γ y (cid:0) r − r (cid:1) ∈ a . Compare that with (5.15) (cid:3)
The family of representations E y,σ,r is compatible with parabolic induction, undera mild condition that ( σ, r ) is not a zero of a certain polynomial function (cid:15) [Lus6,Corollary 1.18]. Namely, let Q be a standard Levi subgroup of G containing L andsuppose that { y, σ } ⊂ Lie( Q ). When (cid:15) ( σ, r ) (cid:54) = 0 (or r = 0, see [AMS2, TheoremA.2]), there is a canonical isomorphism(5.16) ind H ( G,L, L , r ) H ( Q,L, L , r ) E Qy,σ,r → E y,σ,r . The Iwahori–Matsumoto involution commutes with parabolic induction, so (5.16)also holds for the family of representations (cid:101) E y,σ ,r,ρ . With that and Theorem 5.6 onecan show that (at least when (cid:60) ( r ) > (cid:101) E y,σ ,r,ρ is a standard module in the sense ofParagraph 3.5, see [AMS2, Proposition A.3]. (This refers to the last arXiv-versionof [AMS2], in which an appendix was added to deal with a mistake in the publishedversion.) The argument for standardness is analogous to what we sketched for K t,u,ρ around (5.4). FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 57
Example 5.8.
We illustrate the material in this section with an example of rank 1.Let G = Sp ( C ) , L = Sp ( C ) × GL ( C ) and v regular nilpotent in l . The local systemon C Lv corresponding to the sign representation of π ( Z L ( v )) = π ( Z ( L )) = {± } is cuspidal. The root system with respect to T = Z ( L ) ◦ ∼ = C × is R ( G, T ) = {± α, ± α } , of type BC . Its Weyl group is W L = { , s α } , with s α acting on T by inversion. The parameter function by determined by L satisfies k ( ± α ) = 3 and k ( ± α ) = 6. The associated graded Hecke algebra is H ( G, L, L , r ) = H ( t , W L , k, r ) . The irreducible representations on which r acts by a fixed r ∈ C × were classified inExample 3.34 and for r = 0 in Paragraph 2.2.Let us analyse the possibilities for the geometric parameters. It turns out thatthe condition on ρ can only be met for nilpotent elements y in two adjoint orbits in g : the orbits of v and of v + v (cid:48) , where v (cid:48) is regular nilpotent in Z ( l der ) ∼ = sp ( C ).The cuspidal support condition becomes that the subgroup of π ( Z G ( σ , y )) comingfrom π ( Z ( L )) must act via the sign representation.We may assume that C v (cid:48) is stable under the adjoint action of T . A complete listof representatives for the G -conjugacy classes of parameters for H ( G, L, L , r ) is: σ r diag( r, r, − r, − r ) σ = 0 , r = 0 diag( σ, r, − r, − σ ) , (cid:60) ( σ ) ≥ y v + v (cid:48) v vπ ( Z G ( σ , y )) Z (Sp ( C ) ) Z (Sp ( C )) Z (Sp ( C )) ρ sign (cid:2) triv sign sign E y,σ r ,r,ρ triv St ind H ( G,L, L , r ) O ( t ) ⊗ C C [ r ] ( C diag( − σ,r, − r,σ ) ,r ) (cid:101) E y,σ r ,r,ρ St triv ind H ( G,L, L , r ) O ( t ) ⊗ C C [ r ] ( C diag( σ,r, − r, − σ ) ,r )In the last column we exclude the case σ = 0 , r = 0. For almost all σ ∈ C ,standard module ind H ( G,L, L , r ) O ( t ) ⊗ C C [ r ] ( C diag( σ,r, − r, − σ ) ,r ) is irreducible, and hence equal toboth M y,σ r ,r,ρ and (cid:102) M y,σ r ,r,ρ . The exceptions are σ = ± r , then M y, diag( r,r, − r, − r ) ,r,ρ = St and (cid:102) M y, diag( r,r, − r, − r ) ,r,ρ = triv . Affine Hecke algebras from cuspidal local systems.
In the previous paragraph we associated a graded Hecke algebra to a cuspidallocal system on a nilpotent orbit in a Levi algebra of g . With a process that is moreor less inverse to the reduction theorems in Paragraph 3.4, we can glue a suitablefamily of such algebras into one affine Hecke algebra.Concretely, let G, P, L, v, T, L be as before. Let R ( G, T ) ∨ red ⊂ X ∗ ( T ) be the dualof the reduced root system R ( G, T ) red ⊂ X ∗ ( T ), and consider the root datum R ( G, T ) = ( R ( G, T ) red , X ∗ ( T ) , R ( G, T ) ∨ red , X ∗ ( T ) , ∆ L ) . There are unique parameter functions λ, λ ∗ : R ( G, T ) red → Z ≥ that meet therequirements sketched above, see [AMS3, Proposition 2.1 and (28)]. With those, wedefine(5.17) H ( G, L, L ) = H ( R ( G, T ) , λ, λ ∗ , q ) . This is slightly different from [AMS3, § z instead of q / . To compensate for the square root ( q / versus q ) wecould replace λ, λ ∗ by 2 λ, λ ∗ –but that is not necessary, since we are at liberty tochoose q ∈ R > as we like. When λ, λ ∗ : R → R arise from a cuspidal local system as in (5.17), we call themgeometric parameter functions for R . As we may choose any q ∈ R > , λ, λ ∗ may bescaled by any nonzero real factor, and remain geometric.For a unitary t ∈ T un , we saw in Corollary 3.30 that there is an equivalence ofcategoriesMod f,W L t exp( a ) (cid:0) H ( G, L, L ) (cid:1) ∼ = Mod f, a (cid:0) H ( t , W ( R t ) , k t ) (cid:111) Γ t (cid:1) . Let (cid:102) Z G ( t ) be the subgroup of G generated by Z G ( t ) and the root subgroups U α with s α ( t ) = t . By [AMS3, Theorems 2.5 and 2.9] we may identify H ( t , W ( R t ) , k t ) (cid:111) Γ t = H ( (cid:102) Z G ( t ) , L, L , r ) / ( r − log( q ) / . Here (cid:102) Z G ( t ) has component group Γ t , so it can be disconnected. In that case thesegraded Hecke algebras are a little more general than in the previous paragraph, butthat does not matter much. Example 5.9.
We continue Example 5.8. We identity T with C × by means of themap diag( z, , , z − ) (cid:55)→ z . For t ∈ T un \ { , − } , Z G ( t ) = (cid:102) Z G ( t ) = L and H ( (cid:102) Z G ( t ) , L, L , r ) = H ( T, T, triv , r ) = O ( t ) ⊗ C C [ r ] . The most interesting element of T un is − − , , , − Z G ( −
1) = L but (cid:102) Z G ( −
1) = Sp ( C ) × Sp ( C ). Now the root system is R ( (cid:102) Z G ( − , T ) = {± α } ,with parameter k − ( ± α ) = 2. The associated graded Hecke algebra is H ( (cid:102) Z G ( − , L, L , r ) = H ( t , W L , k − , r ) . These data suffice to determine the parameters for H ( G, L, L ), they are λ ( α ) = k ( α ) + k − (2 α ) / λ ∗ ( α ) = k ( α ) − k − (2 α ) / . Thus H ( G, L, L ) = H ( R ( G, T ) , λ, λ ∗ , q ), an affine Hecke algebra of type (cid:101) A withunequal parameters.With Theorems 3.28 and 3.29 we can reduce the classification of Irr( H ( G, L, L ))to Theorem 5.6. For better additional benefits, we prefer to use the modules (cid:101) E y,σ ,r,ρ and (cid:102) M y,σ ,r,ρ . In this context it is more natural to use data from G that from g . Thecorrect parameters turn out to be a variation on Kazhdan–Lusztig triples, namelytriples ( s, u, ρ ) where • s ∈ G is semisimple, • u ∈ Z G ( s ) is unipotent, • ρ ∈ Irr (cid:0) π ( Z G ( s, u )) (cid:1) such that the cuspidal support Ψ Z G ( s ) ( u, ρ ) is( L, exp( C Lv ) , exp ∗ ( L )) modulo G -conjugacy.When Z G ( s ) is disconnected, we have to use a generalization of the cuspidal supportmap, defined in [AMS1, § G , we mayalways assume that s ∈ T . Let E s,u,ρ (resp. M s,u,ρ ) be the H ( G, L, L )-moduleobtained from (cid:101) E log u, log | s | , log( q ) / ,ρ ∈ Mod (cid:0) H ( (cid:102) Z G ( s | s | − ) , L, L , r ) / ( r − log( q ) / (cid:1) (resp. (cid:102) M log u, log | s | , log( q ) / ,ρ ) via Theorems 3.29 and 3.28, with respect to s | s | − ∈ T . Theorem 5.10. [AMS3, Theorem 2.11]
Let q ∈ R > and consider triples ( s, u, ρ ) as above. FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 59 ( a ) The maps ( s, u, ρ ) (cid:55)→ E s,u,ρ (cid:55)→ M s,u,ρ provide canonical bijections { triples as above } /G −→ { standard H ( G, L, L ) -modules } −→ Irr( H ( G, L, L )) . ( b ) Suppose that s ∈ T and let γ u : SL ( C ) → Z G ( s ) be an algebraic homomor-phism with γ u (( )) = u . Then E s,u,ρ and M s,u,ρ admit the central character W L sγ u ( (cid:16) q / q − / (cid:17) ) . ( c ) Suppose that q ≥ . The following are equivalent: • E s,u,ρ is tempered, • M s,u,ρ is tempered, • s lies in a compact subgroup of G . ( d ) Suppose that q > . Then M s,u,ρ is essentially discrete series if and only if u isdistinguished unipotent in G (that is, not contained in any proper Levi subgroupof G ). Notice that q = 1 is allowed here. For q = 1, Theorem 5.10 provides a parametriza-tion of Irr( X ∗ ( T ) (cid:111) W ( G, T )) with G -association classes of triples ( s, u, ρ ) as above.That can be regarded as an affine version of the generalized Springer correspondencefrom [Lus1]. Example 5.11.
We work out the parametrization from Theorem 5.10 for (
G, L, L )as in Examples 5.8 and 5.9. Here H ( G, L, L ) is of type (cid:102) A , with parameters λ ( α ) = 4and λ ∗ ( α ) = 2. With the example at the end of Paragraph 5.2 at hand, it is easy todetermine all geometric parameters ( s, u, ρ ) for H ( G, L, L ), and the relevant modulescan be found by applying Theorems 3.29 and 3.28. s − − , , , − s ∈ T ∼ = C × , | s | ≥ u exp( v + v (cid:48) ) exp( v + v (cid:48) ) exp( v ) π ( Z G ( s, u )) Z (Sp ( C ) ) Z (Sp ( C ) ) Z (Sp ( C )) ρ sign (cid:2) triv sign (cid:2) triv sign E s,u,ρ St π ( − , St) ind H ( G,L, L ) C [ X ∗ ( T )] ( C s )For almost all s ∈ T the standard module ind H ( G,L, L ) C [ X ∗ ( T )] ( C s ) is irreducible, and henceequal to M s, , triv . The exceptions are M diag( q / , , , q − / ) , , triv = triv and M diag( − q / , , , − q − / ) , , triv = π ( − , triv) . A comparison with Paragraph 2.3 shows that we indeed found every irreducible H ( G, L, L )-representation once in this way.We have associated to every complex reductive group G a family of affine Heckealgebras, one for every cuspidal local system on a nilpotent orbit for a Levi subgroupof G . Every nilpotent orbit admits only a few inequivalent cuspidal local systems,and G -conjugate data ( L, C Lv , L ) yield isomorphic Hecke algebras. Thus we have afinite family of affine Hecke algebras associated to G .The simplest member of this family arises when L = T, v = 0 and L is trivial.Then the graded Hecke algebras H ( (cid:102) Z G ( t ) , T, L = triv) have parameters k ( α ) = 2for all α ∈ R ( G, T ). When we specialize r to log( q ) /
2, (3.18) shows that λ ( α ) = λ ∗ ( α ) = 1 for all α ∈ R ( G, T ). In other words: H ( G, T, L = triv) = H ( R ( G, T ) , q ) . For this algebra the cuspidal support condition on the triples ( s, u, ρ ) reduces tothe condition on ρ in a Kazhdan–Lusztig triple for ( G, H ( G, T, L = triv)-modules inTheorem 5.10 coincides with the Kazhdan–Lusztig paramerization from Theorem5.1, modified as in Corollary 5.2. That is less obvious than it might seem though,the twist with the Iwahori–Matsumoto involution in (5.14) is necessary to achievethe agreement.6. Comparison between different q -parameters The aim of this section is a canonical bijection between the set of irreduciblerepresentations of an affine Hecke algebra with arbitrary parameters q s ∈ R ≥ , andthe set of irreducible representations of the same algebra with parameters q s = 1.This will be achieved in several steps of increasing generality.6.1. W -types of irreducible tempered representations. Consider any graded Hecke algebra H = H ( t , W, k ). The group algebra C [ W ] isembedded in H , so every H -representation can be restricted to a W -representation.For k = 0, the isomorphism(6.1) H ( t , W, / ( t ∗ ) ∼ = C [ W ]shows that a representation on which O ( t ) acts via evaluation at 0 ∈ t is the same asa C [ W ]-representation. From Example 3.7 we know that the irreducible temperedrepresentations of H ( R ,
1) with central character in exp( a ) are precisely the irre-ducible representations which admit the O ( T )-character 1 ∈ T . Via Corollary 3.30this implies that the irreducible representations of (6.1) are precisely the irreducibletempered H ( t , W, a /W .That and the results of Paragraph 3.4 indicate that we should focus on H -representations with O ( t )-weights in a = X ∗ ( T ) ⊗ Z R . We say that those havereal weights. LetIrr ( H ( t , W, k )) = { π ∈ Mod f, a ( H ( t , W, k )) : π is irreducible and tempered } be the set of irreducible tempered representations with real central character. Theabove says that Irr ( H ( t , W, W ). Since H ( t , W, k ) isa deformation of H ( t , W, H ( t , W, k ). On closer inspection the parameters k ( α ) interact with the notion oftemperedness, and it is natural to require that k takes real values.For later applications to affine Hecke algebras, it will pay off to increase ourgenerality. Let Γ be a finite group which acts on ( a ∗ , z ∗ , R, ∆). That is, Γ acts R -linearly on a ∗ , and that action stabilizes R, ∆ and the decomposition a ∗ = R R ⊕ z ∗ .Suppose further that k : R → R is constant on Γ-orbits. Then Γ acts on H ( t , W, k )by the algebra automorphisms ξw (cid:55)→ γ ( ξ )( γwγ − ) ξ ∈ t ∗ , w ∈ W. The crossed product algebra H ( t , W, k ) (cid:111) Γ =
H(cid:111)
Γ is of the kind already encounteredin Corollary 3.30. The Γ-action on H preserves all the available structure, so all theusual notions for H also make sense for H (cid:111)
Γ.We denote the restriction of any
H(cid:111)
Γ-representation π to the subalgebra C [ W (cid:111) Γ]by Res W (cid:111) Γ ( π ). An initial result in the direction sketched above is: FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 61
Theorem 6.1. [Sol2, Theorem 6.5]Irr ( H(cid:111) Γ) and Irr( W (cid:111) Γ) have the same cardinality, and the set Res W (cid:111) Γ (Irr ( H(cid:111)
Γ)) is linearly independent in the representation ring of W (cid:111) Γ . In particular, it is possible to choose a bijection Irr ( H (cid:111) Γ) → Irr( W (cid:111) Γ) suchthat the image of π ∈ Irr ( H (cid:111)
Γ) is always a constituent of Res W (cid:111) Γ ( π ). Wewill establish a much more precise version of Theorem 6.1, for almost all positiveparameter functions k . Theorem 6.2.
Let k : R → R ≥ be a Γ -invariant parameter function whose restric-tion to any type F component of R is geometric or has k ( α ) = 0 for a root α inthat component. ( a ) The set
Res W (cid:111) Γ (Irr ( H (cid:111)
Γ)) is a Z -basis of Z Irr( W (cid:111) Γ) . ( b ) There exist total orders on
Irr ( H (cid:111) Γ) and on Irr( W (cid:111) Γ) such that the matrixof the Z -linear map Res W (cid:111) Γ : Z Irr ( H (cid:111) Γ) → Z Irr( W (cid:111) Γ) is upper triangular and unipotent. ( c ) There exists a unique bijection ζ H(cid:111) Γ : Irr ( H (cid:111) Γ) → Irr( W (cid:111) Γ) such that, for any π ∈ Irr ( H (cid:111) Γ) , ζ H(cid:111) Γ ( π ) occurs in Res W (cid:111) Γ ( π ) . Part (a) is known from [Sol6, Proposition 1.7] and part (c) is a direct consequenceof part (b). Part (b) was already conjectured by Slooten [Slo1, § R = F aregiven by ( k ( α ) , k ( β )) equal to( k, k ) , (2 k, k ) , ( k, k ) or (4 k, k )for any k ∈ C × , where α is a short root and β is a long root. We expect thatTheorem 6.2.b also holds for non-geometric parameter functions k : F → R > , butfound it too cumbersome to check.The discussion after (6.1) shows that the theorem is trivial for k = 0, so we assumefrom now on that k (cid:54) = 0. Our proof of Theorem 6.2 will occupy the entire paragraph. Lemma 6.3.
Theorem 6.2 holds for H ( t , W, k ) with k : R → R > geometric.Proof. The only irreducible tempered representation with real central character of H ( t , R = ∅ , k ) = O ( t ) is C , so the result is trivially true for that algebra. In viewof the decomposition of H according to the irreducible components of R (1.25), wemay assume that R is irreducible and spans a ∗ .The algebra isomorphisms (5.8)–(5.10) are the identity on O ( t ), so they preservethe set Irr ( H ). The same holds for m z from (1.22), when z ∈ R > . Therefore wemay just as well suppose that k is one of the parameter functions in the table (5.7),and that H = H ( G, L, L , r ) / ( r − r ) for some r ∈ R > . By Theorem 5.7 Irr ( H ) consists of the representations (cid:102) M y, ,r,ρ with y ∈ g nilpotent, σ = 0, ρ ∈ Irr (cid:0) π ( Z G ( y )) (cid:1) and Ψ G ( y, ρ ) = ( L, C Lv , L ) up to G -conjugacy. ByLemma 3.33 all the irreducible constituents of (cid:101) E y, ,r,ρ are tempered and have centralcharacter in a /W . For r = 0 all these representations admit the O ( t )-character 0,so they can be identified with W -representations via (6.1). Recall from (5.11) that Res W (cid:101) E y, ,r,ρ = Res W (cid:101) E y, , ,ρ . Theorem 5.6.d for r = 0 says that all constituents of (cid:101) E y, , ,ρ different from (cid:102) M y, , ,ρ are of the form (cid:102) M y (cid:48) , , ,ρ (cid:48) with dim C Gy < dim C Gy (cid:48) . The numbers dim C Gy define apartial order on the set of eligible pairs ( y, ρ ), considered modulo G -conjugation.Refine that to a total order. We transfer that to a total ordering on Irr ( H ) (resp.Irr( W )) via ( y, ρ ) (cid:55)→ (cid:102) M y, ,r,ρ (resp. (cid:102) M y, , ,ρ ). With respect to these orders, thematrix of Res W : Z Irr ( H ) → Z Irr( W )is unipotent and upper triangular. It follows that ζ H : (cid:102) M y, ,r,ρ (cid:55)→ (cid:102) M y, , ,ρ is the unique map Irr ( H ) → Irr( W ) with the required properties. (cid:3) We remark that Lemma 6.3 with respect to Lusztig’s alternative version of tem-peredness was proven in [Ciu, § R is irreducible and all roots have the same length, k is determined bythe single number k ( α ) >
0, and it is geometric. So we just dealt with R of type A n , D n , E , E or E . For R of type B n , C n , F or G , let k be the k -parameter ofa long root and k the k -parameter of a short root. We will consider the algebras H ( t , W ( R ) , k ) with k = 0 or k = 0 in and after (6.11).That settles R = F for the moment, because we excluded non-geometric strictlypositive parameters. For R = G with strictly positive k , Theorem 6.2 was provenin [Slo1, § H with R irreducible, that leaves type B n or C n . In view of theisomorphism (5.8), it suffices to consider R = B n . Lemma 6.3 proves Theorem 6.2for the following geometric k (and any p ∈ Z > ): k /k g l / sp n Cartan p so p +2 n so p ⊕ C n p + 1 / sp p ( p +1)+2 n sp p ( p +1) ⊕ C n p − / sp p (2 p − n sl n ⊕ C n p + 1 / sp p +1)+4 n sl n ⊕ C n Recall from (4.8) that a strictly positive k is generic if (cid:81) n − j =1 ( jk − k ) is nonzero.In particular we already covered all strictly positive non-generic parameters for B n . Lemma 6.4.
Theorem 6.2 holds for H ( C n , W ( B n ) , k ) when k /k ∈ ( p − / , p ) ∪ ( p, p + 1 / for a p ∈ Z > .Proof. We only consider k /k ∈ ( p − / , p ), the case k /k ∈ ( p, p + 1 /
2) is com-pletely analogous. Define k (cid:48) by k (cid:48) = 1 and k (cid:48) = p − / k with k /k ∈ ( p − / , p ) are generic for B n and for all itsparabolic root subsystems. Hence the residual subspaces of a for k (Definition 4.6)are canonically in bijection with those for k (cid:48) . More precisely, every residual subspacehas coordinates that are linear functions of k , see Proposition 4.10 and (4.7). If sucha linear function gives a residual subspace for k (cid:48) , then it also gives a residual subspacefor k , and conversely. From Theorem 4.13 we obtain a canonical bijection between FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 63 the sets of irreducible discrete series representations of H P ( C n , W ( B n ) , k (cid:48) ) and of H P ( C n , W ( B n ) , k ), say δ (cid:48) (cid:55)→ δ , where the image depends continuously on k .By Lemma 3.33 and (4.5), Irr ( H ( C n , W ( B n ) , k )) consists of the representations π ( P, δ, , ρ ) with ρ ∈ Irr( C [ R P,δ, , (cid:92) P,δ, ]). Recall from Paragraph 4.1 that the an-alytic R-group R P,δ,λ is defined in terms of the functions ˜ c α ( λ ). Since k is onlyallowed to vary among generic parameters, the pole order of ˜ c α | a P ∗ at λ = 0 doesnot depend on k . Hence R P,δ, does not depend on k either. The intertwiningoperators π ( w, P, δ,
0) ( w ∈ R P,δ, ) that span the twisted group algebra can be con-structed so that they depend continuously on k . Then C [ R P,δ, , (cid:92) P,δ, ]) becomes acontinuous family of finite dimensional semisimple C -algebras. Such algebras cannotbe deformed continuously, so the family is isomorphic to a constant family. Thatprovides a canonical bijectionIrr( C [ R P,δ (cid:48) , , (cid:92) P,δ (cid:48) , ]) → Irr( C [ R P,δ, , (cid:92) P,δ, ]) , We plug it into (4.5) and we obtain a bijection(6.2) Irr (cid:0) H ( C n , W ( B n ) , k (cid:48) ) (cid:1) → Irr (cid:0) H ( C n , W ( B n ) , k ) (cid:1) , where the image depends continuously on k . Finite dimensional representations ofthe finite group W do not admit continuous deformations, so (6.2) preserves W -types. Knowing that, Theorem 6.2 for H ( C n , W ( B n ) , k (cid:48) ), as shown in Lemma 6.3,immediately implies Theorem 6.2 for H ( C n , W ( B n ) , k ). (cid:3) Now we involve the group Γ that acts on H via automorphisms of ( a ∗ , z ∗ , R, ∆). Lemma 6.5.
Let H be one of the graded Hecke algebras for which we already provedTheorem 6.2. We can choose the total orders on Irr ( H ) and on Irr( W ) such that,for any π ∈ Irr ( H ) and any irreducible constituent π (cid:48) of Res W ( π ) different from ζ H ( π ) : γ ∗ ( π (cid:48) ) > ζ H ( π ) for all γ ∈ Γ .Proof. First we assume that R is irreducible and spans a ∗ .We consider a geometric k and we revisit the proof of Lemma 6.3. Replacing H by an isomorphic algebra, we may assume that k comes from table (5.7). Inspectionof the table shows that every automorphism of ( R, ∆) can be lifted to an automor-phism of ( G, T ) (Recall that there are no automorphisms of ( R, ∆) when R has type B d , BC d , G or F , apart from the identity.) Hence the functionIrr( W ) → R : (cid:102) M y, , ,ρ (cid:55)→ dim C Gy is Γ-invariant. We replace this function by a function f R,k : Irr( W ) → R , whoseimages differ only slightly from dim C Gy and which induces the total order on Irr( W )claimed in Theorem 6.2.b and exhibited in the proof of Lemma 6.3. Via ζ H , we alsoregard f R,k as a function Irr ( H ). Then Theorem 6.2.c implies that every constituentof Res W ( (cid:102) M y, ,r,ρ ) different from (cid:102) M y, , ,ρ is isomorphic to a (cid:102) M y (cid:48) , , ,ρ (cid:48) with(6.3) f R,k (cid:0) γ ∗ ( (cid:102) M y (cid:48) , , ,ρ (cid:48) ) (cid:1) > f R,k ( (cid:102) M y, , ,ρ ) for all γ ∈ Γ . When R = G and k is not geometric, we can use the analysis from [Slo1, § f R,k : Irr( W ) → R with analogous properties. For other non-geometric k , we may assume that R = B n (recall we imposed that k is geometricfor R = F ). Then k is one of the parameter functions considered in Lemma 6.4. In the proof of that lemma we saw that k can be deformed continuously to a geometricparameter function k (cid:48) , while staying generic. That led to a canonical bijection(6.4) Irr ( H ) → Irr ( H ( C n , W ( B n ) , k (cid:48) )) , which preserves W -types. We define f R,k to be the composition of f R,k (cid:48) with thatbijection, and we transfer it to a function on Irr( W ) via ζ H . The bijection (6.4)and ζ H are Γ-equivariant, if nothing else because ( B n , ∆) does not admit nontrivialautomorphisms. Hence all the properties of f R,k (cid:48) transfer to f R,k .So far we proved the lemma in all cases where R is irreducible and we alreadyhad Theorem 6.2, and we made Theorem 6.2.b more explicit by associating the totalorder to a real-valued function f R,k . For a general R we use the decomposition (1.25)of H . It provides a natural bijectionIrr ( H ( t , W ( R ) , k ) × · · · × Irr ( H ( t d , W ( R d ) , k )) → Irr ( H ( t , W ( R ) , k )( V , . . . , V d ) (cid:55)→ V ⊗ · · · ⊗ V d ⊗ C where { C } = Irr( O ( z ∗ )). We may assume that all values of the f R i ,k constructedabove are algebraically independent and differ from an integer by at most (8 d ) − (ifnot, we can adjust them a bit). Now we define f R,k : Irr ( H ) → R by f R,k ( V ⊗ · · · V d ⊗ C ) = (cid:88) di =1 f R i ,k ( V i ) . and we order Irr ( H ) accordingly. Via ζ H we transfer f R,k and the total order toIrr( W ). Let π (cid:48) be a constituent of Res W ( V ⊗ · · · ⊗ V d ⊗ C ) different from ζ H ( V ⊗· · · ⊗ V d ⊗ C ) and let π (cid:48) i ∈ Irr( W ( R i )) , i = 1 , . . . , d be its tensor components. Atleast one of the π (cid:48) i is not isomorphic to ζ H ( t i ,W ( R i ) ,k ) ( V i ). By (6.3) and its analoguesfor other irreducible R : (cid:3) (6.5) f R,k ( γ ∗ ( π (cid:48) )) > f R,k ( π ) + 1 − d/ d = f R,k ( ζ H ( π )) + 3 / γ ∈ Γ . Before we continue, we quickly recall how Clifford theory relates the irreduciblerepresentations of H and of H (cid:111)
Γ. For ( π, V π ) ∈ Irr( H )) we writeΓ π = { γ ∈ Γ : γ ∗ ( π ) ∼ = π } . For every γ ∈ Γ π we pick a nonzero intertwining operator I γ : π → γ ∗ ( π ). BySchur’s lemma I γ is unique up to scalars, so there exist (cid:92) π ∈ C × such that(6.6) I γγ (cid:48) = (cid:92) π ( γ, γ (cid:48) ) I γ I γ (cid:48) for all γ, γ (cid:48) ∈ Γ π . Then (cid:92) ± π is a 2-cocycle Γ π × Γ π → C × and the twisted group algebra C [Γ π , (cid:92) − π ]acts on V π via the I γ . For every representation ( σ, V σ ) of C [Γ π , (cid:92) π ], the vector space V π ⊗ C V σ becomes a representation of H (cid:111) Γ π by hγ · ( v π ⊗ v σ ) = π ( h ) I γ ( v π ) ⊗ σ ( γ ) v σ . When σ is irreducible, V π ⊗ V σ is also irreducible. Moreover π (cid:111) σ := ind H(cid:111) Γ H(cid:111) Γ π ( V π ⊗ V σ )is an irreducible H(cid:111)
Γ-representation. By [RaRa, Appendix] every irreducible
H(cid:111)
Γ-representation is of the form π (cid:111) σ , for a pair ( π, σ ) that is unique up to the Γ-action.The restriction of π (cid:111) σ to H has constituents γ ∗ ( π ) for γ ∈ Γ / Γ π , each appearingwith multiplicity dim V σ . Since Γ stabilizes a , π (cid:111) σ has all O ( t )-weights in a ifand only if that holds for π . As Γ stabilizes ∆, it preserves temperedness of H -representations. Consequently π (cid:111) σ is tempered if and only if π is tempered. In FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 65 particular Irr ( H (cid:111)
Γ) consists of the representations π (cid:111) σ with π ∈ Irr ( H ) and σ ∈ Irr( C [Γ π , (cid:92) π ]). Lemma 6.6.
Let H be one of the graded Hecke algebras for which we already provedTheorem 6.2 and Lemma 6.5. Then Theorem 6.2 holds for H (cid:111) Γ .Proof. The same Clifford theory as above can also be used to relate the irreduciblerepresentations of W and of W (cid:111) Γ. Recall that Γ acts on W by automorphisms ofthe Coxeter system ( W, S ). In this setting it is known from [ABPS1, Proposition4.3] that the 2-cocycle (cid:92) π W associated to any π W ∈ Irr( W ) is trivial in H (Γ π W , C × ).Hence we can find I γ for π W such that(6.7) Γ π W → Aut C ( V π W ) : γ (cid:55)→ I γ is a group homomorphism. Then Irr( W (cid:111) W ) can be parametrized by Γ-orbits ofpairs ( π W , σ W ) with π W ∈ Irr( W ) and σ W ∈ Irr(Γ π W ).We consider any π ∈ Irr ( H ). By the uniqueness in Theorem 6.2.c, ζ H is Γ-equivariant and Γ π = Γ ζ H ( π ) . The intertwiners I γ : π → γ ∗ ( π ) also qualify asintertwiners I γ : ζ H ( π ) → γ ∗ ( ζ H ( π )), because ζ H ( π ) ⊂ Res W ( π ). Therefore we canspecify a unique I γ : π → γ ∗ ( π ) by the requirement that its restriction to ζ H ( π )equals the I γ from (6.7). ThenΓ π → Aut C ( V π ) : γ (cid:55)→ I γ is a group homomorphism. Now Clifford theory parametrizes Irr ( H(cid:111)
Γ) via Γ-orbitsof pairs ( π, σ ) with π ∈ Irr ( H ) and σ ∈ Irr(Γ π ). In particular we obtain a bijection(which will be ζ H(cid:111) Γ )(6.8) Irr ( H (cid:111) Γ) → Irr( W (cid:111) Γ) : π (cid:111) σ (cid:55)→ ζ H (cid:111) σ. Let f R,k : Irr ( H ) → R be as in the proof of Lemma 6.5. We define(6.9) f R,k ( π (cid:111) σ ) = min { f R,k ( γ ∗ ( π )) : γ ∈ Γ } . Notice that the irreducible W (cid:111) Γ-representation ζ H (cid:111) σ appears in Res W (cid:111) Γ ( π (cid:111) σ ).For any other irreducible constituent π (cid:48) W (cid:111) Γ of Res W (cid:111) Γ ( π (cid:111) σ ), every irreducible W -subrepresentations of π (cid:48) W (cid:111) Γ is contained in Res W ( γ ∗ ( π )) for some γ ∈ Γ. Since ζ H ( π )appears with multiplicity one in Res W ( π ), the subspace ζ H ( π ) (cid:111) σ of Res W (cid:111) Γ ( π (cid:111) σ )exhausts the W -subrepresentations Res W (cid:0) γ ∗ ( ζ H ( π )) (cid:1) in Res W (cid:111) Γ ( π (cid:111) σ ). Hence π (cid:48) W (cid:111) Γ has W -constituents γ ∗ ( π (cid:48) W ) with π (cid:48) W ⊂ Res W ( π ) but π (cid:48) W (cid:54)∼ = ζ H ( π ). Lemma6.5 and (6.5) say that(6.10) f R,k ( γ ∗ ( π (cid:48) W )) > / f R,k ( ζ H ( π )) for all γ ∈ Γ . Take a total order on Irr ( H (cid:111)
Γ) that refines the partial order defined by f R,k . Wetransfer f R,k and this total order to Irr( W (cid:111) Γ) via the bijection (6.8). Then theabove verifies parts (a) and (b) of Theorem 6.2 for
H (cid:111)
Γ. It follows this there is aunique ζ H(cid:111) Γ that fulfills the requirements, namely (6.8) (cid:3) Notice that in Lemma 6.4 we did not allow k /k ∈ (0 , / H ( C n , W ( B n ) , k ) = H ( C n , W ( D n ) , k ) (cid:111) (cid:104) s e n (cid:105) when k = 0 , H ( C n , W ( B n ) , k ) = H ( C n , W ( A ) n , k ) (cid:111) S n when k = 0 , H ( C , W ( F ) , k ) ∼ = H ( C , W ( D ) , k i ) (cid:111) S when k − i = 0 , H ( C , W ( G ) , k ) ∼ = H ( C , W ( A ) , k i ) (cid:111) S when k − i = 0 . Let us relate some of these cases.
Lemma 6.7.
Let k : B n → R > be a parameter function with k /k ∈ (0 , / .There exists a canonical bijection Irr (cid:0) H ( C n , W ( B n ) , k ) (cid:1) → Irr (cid:0) H ( C n , W ( D n ) , k ) (cid:111) (cid:104) s e n (cid:105) (cid:1) which preserves W ( B n ) -types.Proof. We abbreviate H = H ( C n , W ( B n ) , k ) and H (cid:48) = H ( C n , W ( D n ) , k ) (cid:111) (cid:104) s e n (cid:105) . Asexplained in the proof of Lemma 6.4, for k within the range of parameters consideredin this lemma, Irr ( H ) is essentially independent of k . By varying k continuously,we can reach the algebra H (cid:48) , which however may behave differently. By Cliffordtheory Irr ( H (cid:48) ) consists of representations of the following kinds: (i) ind H (cid:48) H ( C n ,W ( D n ) ,k ) ( π ), where π ∈ Irr ( H ( C n , W ( D n ) , k )) is not equivalent with s ∗ e n ( π ), (ii) V π ⊗ V σ , where π ∈ Irr ( H ( C n , W ( D n ) , k )) is fixed by s ∗ e n and σ ∈ Irr( (cid:104) s e n (cid:105) ) = { triv , sign } .Let us investigate what happens when we deform k = 0 to a positive but verysmall real number. Accordingly we replace π (cid:48) = ind H (cid:48) H ( C n ,W ( D n ) ,k ) ( π ) by π = ind HH ( C n ,W ( D n ) ,k ) ( π ) . The map wf (cid:55)→ wf f ∈ O ( C n ) , w ∈ W ( B n )is a linear bijection H (cid:48) → H , so Res W ( B n ) π (cid:48) = Res W ( B n ) π . (i’) We claim that in case (i) π is still irreducible.As vector spaces V π = V π ⊕ s e n V π . For any nonzero linear subspace V of V π , theirreducibility of π (cid:48) tells us that there exists an h ∈ H (cid:48) such that π (cid:48) ( h ) V (cid:54)⊂ V . For k > H still satisfies π ( h ) V (cid:54)⊂ V . Thisverifies the claim (i’). (ii’) In case (ii), we claim that π is reducible.From Theorem 6.1 we know that(6.12) | Irr ( H ) | = | Irr( W ( B n )) | = | Irr ( H (cid:48) ) | (i) and (ii) provide a way to count the right hand side: • every (cid:104) s e n (cid:105) -orbit of length two in Irr ( H ( C n , W ( D n ) , k )) contributes one, • every s ∗ e n -fixed element of Irr ( H ( C n , W ( D n ) , k )) contributes twoThe restriction of any element of Irr ( H ) to H ( C n , W ( D n ) , k ) has all irreducibleconstituents in Irr ( H ( C n , W ( D n ) , k )). By Frobenius reciprocity, this implies thatit is a constituent of π for some π ∈ Irr ( H ( C n , W ( D n ) , k )).When s ∗ e n ( π ) (cid:54)∼ = π , we saw in (i’) that { π, s ∗ e n ( π ) } contributes just one representa-tion to Irr ( H ). In case s ∗ e n ( π ) ∼ = π , π can be reducible. It has length at most two,because that is its length as H ( C n , W ( D n ) , k )-module. When π would contributeonly one representation to Irr ( H ), the sum of the contributions from the cases (i’)and (ii’) would be strictly smaller than the corresponding sum of the contributionsfrom (i) and (ii) to Irr( H (cid:48) ). However, that would contradict (6.12). We concludethat (ii’) holds.A π as in (ii’) has length 2, and both its irreducible constituents become isomor-phic to π upon restriction to H ( C n , W ( D n ) , k ). AsRes W ( B n ) ( π ) = ind W ( B n ) W ( D n ) Res W ( D n ) ( π ) , FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 67 the restriction to W ( B n ) of the two constituents of π must be Res W ( D n ) ( π ) ⊗ trivand Res W ( D n ) ( π ) ⊗ sign. Here Res W ( D n ) ( π ) extends to a representation of W ( B n ),while triv and sign are representations of W ( B n ) /W ( D n ). In combination with case(i’) we see that Res W ( B n ) (Irr ( H (cid:48) )) = Res W ( B n ) (Irr ( H )) . Hence there is a unique bijection Irr ( H ) → Irr ( H (cid:48) ) that preserves W -types. (cid:3) Finally, we settle the remaining cases of Theorem 6.2.
Proof.
Lemma 6.6 establishes Theorem 6.2 for the algebras in (6.11). Moreover (6.9)gives us a function f R,k that defines a useful partial order on Irr ( H ) and Irr( W ).With Lemma 6.7 we transfer all that to H ( C n , W ( B n ) , k ) with k /k ∈ (0 , / R is irreducible and spans a ∗ . As noted in theproof of Lemma 6.3, that implies Theorem 6.2 for all H (still with the condition onthe parameters for type F components). We finish the proof by applying Lemmas6.5 and 6.6 another time. (cid:3) Suppose that H = H ( G, L, L , r ) / ( r − r ) for a cuspidal local system L on a nilpotentorbit for L (as in Paragraph 5.2). In terms of Theorem 5.7, ζ H from Theorem 6.2 isjust the map (cid:102) M y, ,r,ρ (cid:55)→ (cid:102) M y, , ,ρ . Here ( y, ρ ) (cid:55)→ (cid:102) M y, , ,ρ is the generalized Springercorrespondence from [Lus1], twisted by the sign character of W . So, for a gradedHecke algebra that can be constructed with equivariant homology, Theorem 6.2recovers a generalized Springer correspondence for W .Let us relax this notion, and call any nice parametrization of Irr( W ) a gener-alized Springer correspondence. Then Theorem 6.2 qualifies as such, and we canregard ζ H : Irr ( H ) → Irr( W ) as a ”generalized Springer correspondence withgraded Hecke algebras”. This point of view has been pursued in [Slo1], whereIrr ( H ( C n , W ( B n ) , k )) has been parametrized with combinatorial data that mimicthe above pairs ( y, ρ ).6.2. A generalized Springer correspondence with affine Hecke algebras.
With Corollary 3.30 we can translate Theorem 6.2 into a statement about alltempered irreducible representations of H = H ( R , λ, λ ∗ , q ), in relation with tem-pered C [ X (cid:111) W ]-representations. We want to generalize that to all irreducible H -representations, at least when q s ≥ s ∈ S aff .Our main tool will be the Langlands classification. We need a version for gradedHecke algebras extended with automorphism groups Γ as in Theorem 6.2. It can beobtained by combining Theorem 3.13 for H with Clifford theory. Given a Langlandsdatum ( P, τ, λ ) for H , let Γ P,τ,λ be its stabilizer in Γ, and recall the 2-cocycle (cid:92)
P,τ,λ from (6.6). We define a Langlands datum for
H (cid:111)
Γ to be a quadruple (
P, τ, λ, ρ ),where • ( P, τ, λ ) is a Langlands datum for H (so τ ∈ Irr( H P ) is tempered and λ ∈ a P ++ + i a P ), • ρ ∈ Irr( C [Γ P,τ,λ , (cid:92)
P,τ,λ ]).To such a quadruple we associate the irreducible H P -representation (( τ ⊗ λ ) ⊗ ρ, V τ ⊗ C V ρ ) and the H (cid:111)
Γ-representation π ( P, τ, λ, ρ ) = ind
H(cid:111) Γ H P (cid:111) Γ P,τ,λ ( τ ⊗ λ ⊗ ρ ) . As a consequence of [Sol4, Corollary 2.2.5] and Paragraph 3.5, we find an extendedLanglands classification:
Corollary 6.8.
Let ( P, τ, λ, ρ ) be a Langlands datum for H (cid:111) Γ . ( a ) The
H (cid:111) Γ -representation π ( P, τ, λ, ρ ) has a unique irreducible quotient, whichwe call L ( P, τ, λ, ρ ) . ( b ) For every irreducible
H (cid:111) Γ -representation π , there exists a Langlands datum ( P (cid:48) , τ (cid:48) , λ (cid:48) , ρ (cid:48) ) , unique up to the canonical Γ -action, such that π ∼ = π ( P (cid:48) , τ (cid:48) , λ (cid:48) , ρ (cid:48) ) . ( c ) L ( P, τ, λ, ρ ) and π ( P, τ, λ, ρ ) are tempered if and onlu if P = ∆ and λ ∈ i a ∆ . In this context we call π ( P, τ, λ, ρ ) a standard
H (cid:111)
Γ-module. When τ has realcentral character, it follows from Theorem 6.2 that the representation Res W P ( τ ) ⊗ λ of H ( t , W P ,
0) = O ( t ) (cid:111) W P has stabilizer Γ P,τ,λ in { γ ∈ Γ : γ ( P ) = P } . The H P -intertwining operators I γ ( γ ∈ Γ P,τ,λ ) from (6.11) are also O ( t ) (cid:111) W P -intertwiningoperators, so (Res W P ( τ ) ⊗ λ ⊗ ρ, V τ ⊗ C V ρ ) is a well-defined representation of H ( t , W P , (cid:111) Γ P,τ,λ = O ( t ) (cid:111) W P Γ P,τ,λ . Its parabolic induction is π ( P, Res W P ( τ ) , λ, ρ ) = ind O ( t ) (cid:111) W Γ O ( t ) (cid:111) W P Γ P,τ,λ (Res W P ( τ ) ⊗ λ ⊗ ρ ) . Of course this
H (cid:111)
Γ-representation may have more than one irreducible quotient,because Res W P ( τ ) usually is reducible. Recall from the proof of Lemma 6.6 that wecan arrange that Γ P,τ,λ → Aut C ( V π ) : γ (cid:55)→ I γ is a group homomorphism. Then (cid:92) P,τ,λ = 1 and ρ becomes simply an irreduciblerepresentation of Γ P,τ,λ . This construction yields a canonical map(6.13) Res O ( t ) (cid:111) W Γ : { standard H (cid:111)
Γ-modules } −→
Mod f ( O ( t ) (cid:111) W Γ) π ( P, τ, λ, ρ ) (cid:55)→ π ( P, Res W P ( τ ) , λ, ρ ) . For λ = 0, this just the restriction map Res W (cid:111) Γ , in combination with (6.1). In termsof (4.4) and (4.5), we can express (6.13) asRes O ( t ) (cid:111) W Γ (cid:0) Hom R P,δ,λ (cid:0) ρ, ind H(cid:111) Γ H P ( δ ⊗ λ ) (cid:1)(cid:1) = Hom R P,δ,λ (cid:0) ρ, ind O ( t ) (cid:111) W Γ O ( t ) (cid:111) W P (cid:0) Res W P ( δ ) ⊗ λ (cid:1)(cid:1) . Here we used [Sol3, Theorem 9.2] to extend the notion of R-groups to
H (cid:111) Γ. Lemma 6.9.
Let k : R → R be a parameter function as in Theorem 6.2, so that inparticular Theorem 6.2.b provides a total order > on Irr( W P ) . Let ( P, τ, λ, ρ ) be aLanglands datum for H (cid:111) Γ .All irreducible constituents of π ( P, Res W P ( τ ) , λ, ρ ) different from π ( P, ζ H P ( τ ) , λ, ρ ) are of the form π ( P, τ (cid:48) W , λ, ρ (cid:48) W ) , where τ (cid:48) W > ζ H P ( τ ) and ( P, τ (cid:48) W , λ, ρ (cid:48) W ) is a Lang-lands datum for H ( t , W, (cid:111) Γ = O ( t ) (cid:111) W Γ .Proof. By Theorem 6.2 every irreducible constituent τ (cid:48) W of Res W P ( τ ) different from ζ H P ( τ ) is strictly larger than ζ H P ( τ ). Although the O ( t ) (cid:111) W P -representation τ (cid:48) W ⊗ λ is irreducible, its stabilizer inΓ P,λ = { γ ∈ Γ : γ ( P ) = P, γ ( λ ) = λ } need not be Γ P,τ,λ . To overcome that, we rather work with Γ
P,λ . Put τ (cid:48) =ind H P (cid:111) Γ P,λ H P (cid:111) Γ P,τ,λ ( τ ⊗ ρ ), so that π ( P, τ, λ, ρ ) = ind
H(cid:111) Γ H P (cid:111) Γ P,λ ( τ (cid:48) ⊗ λ ) . FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 69
Take ρ (cid:48) W ∈ Irr(Γ
P,τ (cid:48) W ,λ ) such that(6.14) ind O ( t ) (cid:111) W P Γ P,λ O ( t ) (cid:111) W P Γ P,τ (cid:48)
W ,λ ( τ (cid:48) W ⊗ ρ (cid:48) W )is a subrepresentation of(6.15) Res W P (cid:111) Γ P,λ ( τ (cid:48) ) = ind O ( t ) (cid:111) W P Γ P,λ O ( t ) × W P Γ P,τ,λ (Res W P ( τ ) ⊗ ρ ) . Then π ( P, τ (cid:48) W , λ, ρ (cid:48) W ) is a subrepresentation of π ( P, Res W P ( τ ) , λ, ρ ), and by Corol-lary 6.8 it is irreducible. With this construction we can obtain any subrepresentationof (6.15) whose W P -constituents are not Γ P,λ -associate to ζ H P ( τ ).Since ζ H P ( τ ) appears with multiplicity one in Res W P ( τ ), ζ H P ( τ ) ⊗ λ ⊗ ρ exhaustsall Γ P,τ,λ -associates of ζ H P ( τ ) in Res W P ( τ ) ⊗ λ ⊗ ρ . Then(6.16) ind O ( t ) (cid:111) W P Γ P,λ O ( t ) × W P Γ P,τ,λ ( ζ H P ( τ ) ⊗ λ ⊗ ρ )exhausts all Γ P,λ -associates of ζ H P ( τ ) in(6.17) ind O ( t ) (cid:111) W P Γ P,λ O ( t ) × W P Γ P,τ,λ (Res W P ( τ ) ⊗ λ ⊗ ρ ) . Consequently (6.16) and the modules (6.14) exhaust the whole of (6.17). Thisremains the case after inducing everything to O ( t ) (cid:111) W Γ.Therefore π ( P, Res W P ( τ ) , λ, ρ ) does not have any other subrepresentations besides π ( P, ζ H P ( τ ) , λ, ρ ) and the π ( P, τ (cid:48) W , λ, ρ (cid:48) W ). (cid:3) From (6.13) we will deduce a map whose image consists of irreducible representa-tions of O ( t ) (cid:111) W Γ. By Corollary 2.1, those are the same as standard O ( t ) (cid:111) W Γ-modules. We call a central character for
H (cid:111)
Γ real if it lies in a /W Γ. Proposition 6.10.
Let
H (cid:111) Γ be as in Theorem 6.2. ( a ) There exists a unique bijection ζ H(cid:111) Γ between: • the set of standard H (cid:111) Γ -modules with real central character, • the set of irreducible O ( t ) (cid:111) W Γ -representations with real central character,such that ζ H(cid:111) Γ ( π ) is always a constituent of Res O ( t ) (cid:111) W Γ ( π ) .For suitable total orders on these two sets, the matrix of (the linear extensionof ) ζ H(cid:111) Γ is the identity while the matrix of Res O ( t ) (cid:111) W Γ is upper triangular andunipotent. ( b ) There exists a natural bijection ζ (cid:48) H(cid:111) Γ from the set of irreducible H(cid:111) Γ -representa-tions with real central character to the analogous set for O ( t ) (cid:111) W Γ , such that: • for tempered representations it coincides with part (a), • the restriction of ζ (cid:48) H(cid:111) Γ ( π ) to C [ W (cid:111) Γ] is a constituent of Res W (cid:111) Γ ( π ) .Proof. (a) By Lemma 6.9, any candidate for such a map must preserve the P and the λ in the Langlands datum of π (from Corollary 6.8). By Lemma 3.3 the conditionon the central character of π means that λ ∈ a P ++ and the ingredient τ lies inIrr ( H P ). Hence we can specialize to a map { ( τ, ρ ) : τ ∈ Irr ( H P ) , ρ ∈ Irr(Γ
P,τ,λ ) } −→{ ( τ W , ρ W ) : τ W ∈ Irr( W P ) , ρ W ∈ Irr(Γ
P,τ W ,λ ) } . Here the left hand side parametrizes Irr ( H P (cid:111) W P,λ ) and the right hand sideparametrizes Irr( W P (cid:111) Γ P,λ ). For fixed (
P, λ ) there is a commutative diagram(6.18) (cid:26) standard
H (cid:111)
Γ-moduleswith real central character (cid:27)
Res O ( t ) (cid:111) W Γ −−−−−−−→ Mod f ( O ( t ) (cid:111) W Γ) ↑ ind H(cid:111) Γ H P (cid:111) Γ P,λ ⊗ λ ↑ ind O ( t ) (cid:111) W Γ O ( t ) (cid:111) W P Γ P,λ ⊗ λ Irr ( H P (cid:111) Γ P,λ ) Res WP (cid:111) Γ P,λ −−−−−−−−→
Mod f ( O ( t P ) (cid:111) W P Γ P,λ )where the vertical arrows send τ (cid:111) ρ to π ( P, τ, λ, ρ ) and τ W (cid:111) ρ W to π ( P, τ W , λ, ρ W ).By Theorem 6.2 for H P (cid:111) Γ P,λ , the matrix of Res W P (cid:111) Γ P,λ (with respect to suitabletotal orders) is upper triangular and unipotent. Hence there is a unique map ζ H(cid:111) Γ that fulfills the requirements for fixed ( P, λ ), namely(6.19) ind
H(cid:111) Γ H P (cid:111) Γ P,λ ( τ (cid:111) ρ ⊗ λ ) (cid:55)→ ind O ( t ) (cid:111) W Γ O ( t ) (cid:111) W P Γ P,λ ( ζ H P (cid:111) Γ P,λ ( τ (cid:111) ρ ) ⊗ λ ) . This translates to(6.20) ζ H(cid:111) Γ π ( P, τ, λ, ρ ) = π ( P, ζ H P ( τ ) , λ, ρ ) . Theorem 6.2 and the commutative diagram (6.18) entail the required properties ofthe matrices of ζ H(cid:111) Γ and Res O ( t ) (cid:111) W Γ .(b) By Corollary 6.8 taking the irreducible quotient of a standard module providesa natural bijection(6.21) { standard H (cid:111)
Γ-modules } −→
Irr(
H (cid:111) Γ) . Define ζ (cid:48) H(cid:111) Γ as the composition of the inverse of (6.21) with ζ H(cid:111) Γ from part (a). ByCorollary 6.8 every tempered irreducible module is also standard, so for temperedrepresentations the properties of ζ H(cid:111) Γ remain valid for ζ (cid:48) H(cid:111) Γ . By (6.19)(6.22) Res W (cid:111) Γ ( ζ H(cid:111) Γ ( π ( P, τ, λ, ρ )) = ind W (cid:111) Γ W P (cid:111) Γ P,λ ( ζ H P (cid:111) Γ P,λ ( τ (cid:111) ρ )) , which is generated by ζ H P (cid:111) Γ P,λ ( τ (cid:111) ρ ). By construction (see [Sol4, § π ( P, τ, λ, ρ ) → L ( P, τ, λ, ρ )is the maximal
H(cid:111)
Γ submodule that intersects the vector space V underlying τ (cid:111) ρ ⊗ λ trivially. In particular (6.23) is injective on V , and that realizes Res W P (cid:111) Γ ( τ (cid:111) ρ )as a C [ W P (cid:111) Γ P,λ ]-subrepresentation of L ( P, τ, λ, ρ ). Then ζ H P (cid:111) Γ P,λ ( τ (cid:111) ρ ) is alsoa W P (cid:111) Γ P,λ -subrepresentation L ( P, τ, λ, ρ ). Hence L ( P, τ, λ, ρ ) contains (6.22) as a W (cid:111) Γ-subrepresentation. (cid:3)
With Corollary 3.30 we will transfer Proposition 6.10 to affine Hecke algebraswith parameters in R ≥ . However, there is one detail that we need to get out of theway first. Let λ, λ ∗ : R → R ≥ be parameter functions. Then q s α = q λ ( α ) and q λ ∗ ( α ) s (cid:48) α belong to R ≥ for all α ∈ R , but λ ( α ) − λ ∗ ( α ) could be <
0. In that case Theorem3.29 could produce a graded Hecke algebra with a negative parameter k u ( α ), towhich we could not apply Proposition 6.10. Lemma 6.11.
Let H be an affine Hecke algebra constructed from a root datum R anda parameter function q : S aff → R ≥ . Then H admits a presentation H ( R , λ, λ ∗ , q ) with λ ( α ) , λ ∗ ( α ) , λ ( α ) − λ ∗ ( α ) ∈ R ≥ for all α ∈ R . FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 71
Proof.
Choose λ, λ ∗ : R → R ≥ as in (1.15), so that H ∼ = H ( R , λ, λ ∗ , q ).When λ ∗ ( α ) > λ ( α ), α must be a short root in a type B n component R i of R and R R ∨ i ∩ Y = C n . Use the presentation from Definition 1.11. Replace the basepoint1 of T = Y ⊗ Z C × by (cid:80) ni =1 e i ⊗ − ∈ T W . This produces a new torus T (cid:48) , and thealgebra H can be presented as H ( T (cid:48) , λ (cid:48) , λ ∗ (cid:48) , q ). Here λ (cid:48) ( α ) = λ ∗ ( α ) , λ ∗ (cid:48) ( α ) = λ ( α )and λ (cid:48) = λ, λ ∗ (cid:48) = λ ∗ on R \ W α . This translates to a Bernstein presentation H ( R , λ (cid:48) , λ ∗ (cid:48) , q ) of H , with λ ∗ (cid:48) ( β ) > λ (cid:48) ( β ) for fewer β ∈ R than before. Repeatingthe procedure, we can achieve λ ( α ) ≥ λ ∗ ( α ) for all α ∈ R . (cid:3) From now on H = H ( R , λ, λ ∗ , q ) is as in Lemma 6.11. Let u ∈ T un . By (3.18)the parameter function k u for H ( t , W ( R u ) , k u ) (cid:111) Γ u takes values in R ≥ . Recall fromLemma 3.31 that Theorems 3.28 and 3.29 are compatible with temperedness andparabolic induction. Let π ( P, τ, λ, ρ ) be a standard module for H ( t , W ( R u ) , k u ) (cid:111) Γ u ,with real central character. Via Theorems 3.28 and 3.29 it corresponds naturally toa standard H -module π ( P (cid:48) , τ (cid:48) , t ) with τ (cid:48) ∈ Irr( H P (cid:48) ) tempered and | t | = exp( λ ). Wedefine Res O ( T ) (cid:111) W : { standard H -modules } −→ Mod f ( O ( T ) (cid:111) W )by commutativity of the following diagram (for central characters in W u exp( a )): { standard H -modules } Res O ( T ) (cid:111) W −−−−−−−→ Mod f ( O ( T ) (cid:111) W ) ↑ Corollary 3.30 ↑ Corollary 3.30 (cid:26) standard H ( t , W ( R u ) , k u ) (cid:111) Γ u − modules with real central character (cid:27) Res O ( t ) (cid:111) W ( Ru )Γ u −−−−−−−−−−−→ Mod f ( O ( t ) (cid:111) W ( R u )Γ u )The map Res O ( T ) (cid:111) W can be made more explicit with (4.3) and Corollary 4.4. Inthose terms(6.24) Res O ( T ) (cid:111) W (cid:0) π ( P, δ, t, ρ ) (cid:1) = Hom C [ R P,δ,t ,(cid:92)
P,δ,t ] (cid:0) ρ, π ( P, Res W P ( δ ) , t ) (cid:1) . For tempered standard H -modules (i.e. with t ∈ T P un ), Res O ( T ) (cid:111) W can really be con-sidered as a restriction, see [Sol4, § H -modules. However, it is not a restriction (along some injectivealgebra homomorphism) because it can happen that π ( P, δ, t, ρ ) is reducible but itsimage (6.24) is irreducible.We do not know how to extend Res O ( T ) (cid:111) W to arbitrary H -representations. Thebest we can do is to define the W -type of any finite dimensional H -representation,in the following way: • By decomposing it as in (3.3), we may assume that all its O ( T )-weights liein a single W -orbit, say in W u exp( a ) with u ∈ T un . • Apply Theorems 3.28 and 3.29 to produce a representation of H ( t , W ( R u ) , k u ) (cid:111) Γ u . • Restrict to C [ W ( R u ) (cid:111) Γ u ] and then induce to C [ W ].Of course this mimics the earlier W -type maps for graded Hecke algebras. In [Sol4, § W -type of an H -representation can also be obtainedvia a continuous deformation of q to 1.We are ready transfer Proposition 6.10 to affine Hecke algebras: Theorem 6.12.
Let H = H ( R , λ, λ ∗ , q ) be an affine Hecke algebra with parameterfunctions λ, λ ∗ : R → R ≥ . Suppose that the restrictions λ i = λ ∗ i to any type F component R i of R satisfy: either λ i is geometric or λ i ( α ) = 0 for an α ∈ R i . ( a ) There exists a unique bijection ζ H : { standard H -modules } −→ Irr( O ( T ) (cid:111) W ) such that ζ H ( π ) is always a constituent of Res O ( T ) (cid:111) W ( π ) .There exists a total order on Irr( O ( T ) (cid:111) W ) such that, if we transfer it via ζ H , the matrix of the Z -linear map Res O ( T ) (cid:111) W : Z { standard H -modules } −→ Z Irr( O ( T ) (cid:111) W ) becomes upper triangular and unipotent. ( b ) There exists a natural bijection ζ (cid:48)H : Irr( H ) −→ Irr( X (cid:111) W ) such that: • for irreducible tempered H -representations it coincides with part (a), • for any π ∈ Irr( H ) , Res W ( ζ (cid:48)H ( π )) is a constituent of the W -type of π . The allowed parameter functions for a type F component of R are( λ ( α ) , λ ( β )) ∈ { ( k, k ) , (2 k, k ) , ( k, k ) , (4 k, k ) , ( k, , (0 , k ) , (0 , } , where k ∈ R > , α ∈ F is a short root and β ∈ F is a long root. Like we mentionedafter Theorem 6.2, we believe that Theorem 6.12 is valid for all parameter functions λ, λ ∗ : R → R ≥ .We point out that the uniqueness/naturality is essential in Theorem 6.12. With-out that condition, it would be much easier to derive it from [Sol4, § Proof.
By Lemma 6.11 and the asserted naturality of ζ H and ζ (cid:48)H , we may assumethat λ ( α ) ≥ λ ∗ ( α ) for all α ∈ R . For any u ∈ T un , the parameter function k u from(3.18) takes values in R ≥ .For any irreducible component R i of R not of type F , we claim that R i does notpossess a root subsystem isomorphic to F . This can be seen with a case-by-caseconsideration of irreducible root systems. To get a root subsystem of type F , R i needs to possess roots of different lengths, so it has type B n , C n , F or G . Therank of G is too low, so R (cid:54)∼ = G . In type B n (resp. C n ) the short (resp. long)roots form a subsystem of type ( A ) n , and that does not contain D . As both thelong and the short roots in F form a root system of type D , we can conclude that B n (cid:54)∼ = R i (cid:54)∼ = C n .Now we apply Corollary 3.30 and reduce the theorem to the graded Hecke algebras H ( t , W ( R u ) , k u ) (cid:111) Γ u , for all u ∈ T un . By the above, the parameter function k u fulfillsthe requirements of Theorem 6.2. Finally, we apply Proposition 6.10. (cid:3) We note that Theorem 6.12 provides a natural bijection between the irreducibleor standard modules of two affine Hecke algebras with the same root datum butdifferent parameters q s ≥ H arises from a cuspidal local system L on a nilpotent orbit for a Levisubgroup L of G , Theorem 6.12 boils down to Theorem 5.10. More precisely, in thiscase(6.25) Res O ( T ) (cid:111) W ( E s,u,ρ ) = E s,u,ρ ,ζ H ( G,L, L ) ( E s,u,ρ ) = ζ (cid:48)H ( G,L, L ) ( M s,u,ρ ) = M s,u,ρ , where the terms on the right are representations of O ( T ) (cid:111) W , the version of H ( G, L, L ) with q = 1. The reasons for (6.25) are: FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 73 • the constructions that led to ζ H are analogous to those behind Theorem 5.10, • for graded Hecke algebras from cuspidal local systems we imposed such com-patibility in the proof of Lemma 6.3.Theorem 5.10 with q = 1 can be considered as a generalized Springer correspon-dence for the (extended) affine Weyl group X (cid:111) Γ = W aff (cid:111) Ω, with geometric data( s, u, ρ ). Consequently Theorem 6.12 can also be regarded as a generalized Springercorrespondence of sorts, where the geometric data have been replaced by standardor irreducible modules of an affine Hecke algebra with (nearly arbitrary) parameters q s ∈ R ≥ . References [Aub] A.-M. Aubert, “Local Langlands and Springer correspondences”, pp. 1–37 in:
Represen-tations of Reductive p-adic Groups, A.-M. Aubert, M. Mishra, A. Roche, S. Spallone (eds.) ,Progress in Mathematics , Birkh¨auser, 2019[ABPS1] A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, “The principal series of p -adicgroups with disconnected centre”, Proc. London Math. Soc. (2017), 798–854[ABPS2] A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, “Conjectures about p -adic groupsand their noncommutative geometry”, Contemp. Math. (2017), 15–51[AMS1] A.-M. Aubert, A. Moussaoui, M. Solleveld “Generalizations of the Springer correspondenceand cuspidal Langlands parameters”, Manus. Math. (2018), 121–192[AMS2] A.-M. Aubert, A. Moussaoui, M. Solleveld, “Graded Hecke algebras for disconnected re-ductive groups”, pp. 23–84 in: Geometric aspects of the trace formula, W. M¨uller, S. W. Shin,N. Templier (eds.) , Simons Symposia, Springer, 2018[AMS3] A.-M. Aubert, A. Moussaoui, M. Solleveld, “Affine Hecke algebras for Langlands parame-ters”, arXiv:1701.03593, 2019[BaMo1] D. Barbasch, A. Moy, “A unitarity criterion for p -adic groups”, Inv. Math. (1989),19–37[BaMo2] D. Barbasch, A. Moy, “Reduction to real infinitesimal character in affine Hecke algebras”,J. Amer. Math. Soc. (1993), 611–635[BeZe] J. Bernstein, A. Zelevinsky, “Representations of the group GL ( n, F ) where F is a localnon-archimedean field”, Usp. Mat. Nauk (1976), 5–70[Bou] N. Bourbaki, Groupes et alg`ebres de Lie. Chapitres IV, V et VI , ´El´ements de math ´matique
XXXIV , Hermann, 1968[BuKu1] C.J. Bushnell, P.C. Kutzko, “The admissible dual of GL(N) via compact open subgroups”,Annals of Mathematics Studies , Princeton University Press, 1993[Car] R.W. Carter,
Finite groups of Lie type. Conjugacy classes and complex characters , Pure andApplied Mathematics (New York), John Wiley & Sons, 1985[Cas] W. Casselman, “Introduction to the theory of admissible representations of p -adic reductivegroups”, preprint, 1995[Che1] I. Cherednik, “A unification of Knizhnik-Zamolodchikov and Dunkl operators via affineHecke algebras”, Invent. Math. (1991), 411–431[Che2] I. Cherednik, Double affine Hecke algebras , London Mathematical Society Lecture NoteSeries , Cambridge University Press, 2005[ChGi] N. Chriss, V. Ginzburg,
Representation theory and complex geometry , Birkh¨auser, 1997[Ciu] D. Ciubotaru, “On unitary unipotent representations of p -adic groups and affine Hecke alge-bras with unequal parameters”, Representation Theory (2008), 453–498[CKK] D. Ciubotaru, M. Kato, S. Kato, “On characters and formal degrees of discrete series ofaffine Hecke algebras of classical types”, Invent. Math. (2012), 589–635[CiTr1] D. Ciubotaru, P.E. Trapa, “Duality between GL ( n, R ) , GL ( n, Q p ), and the degenerate affineHecke algebra for gl ( n )”, Amer. J. Math. (2012), 141–170[CiTr2] D. Ciubotaru, P.E. Trapa, “Functors for unitary representations of classical real groups andaffine Hecke algebras”, Advances in Math. (2011), 1585–1611[CuRe] C.W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras , Pureand Applied Mathematics , John Wiley & Sons, 1962 [DeOp1] P. Delorme, E.M. Opdam, “The Schwartz algebra of an affine Hecke algebra”, J. reineangew. Math. (2008), 59–114[DeOp2] P. Delorme, E.M. Opdam, “Analytic R-groups of affine Hecke algebras”, J. reine angew.Math. (2011), 133–172[Dri] V.G. Drinfeld, “Degenerate affine Hecke algebras and Yangians”, Funktsional. Anal. iPrilozhen. (1986), 69–70[Eve] S. Evens, “The Langlands classification for graded Hecke algebras”, Proc. Amer. Math. Soc. (1996), 1285–1290[GoRo] D. Goldberg, A. Roche, “Hecke algebras and SL n -types”, Proc. London Math. Soc. (2005), 87–131[Gyo] A. Gyoja, “Modular representation theory over a ring of higher dimension with applicationsto Hecke algebras” J. Algebra (1995), 553–572[GyUn] A. Gyoja, K. Uno, “On the semisimplicity of Hecke algebras” J. Math. Soc. Japan (1989), 75–79[GePf] M. Geck, G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras , Lon-don Mathematical Society Monographs New Series , Oxford University Press, 2000[HeOp] G.J. Heckman, E.M. Opdam, “Yang’s system of particles and Hecke algebras”, Ann. ofMath. (1997), 139–173[HiTh] F. Hivert, N. M. Thi´ery, “The Hecke group algebra of a Coxeter group and its representationtheory”, J. Algebra (2009), 2230–2258[HST] F. Hivert, A. Schilling, N.M. Thi´ery, “The biHecke monoid of a finite Coxeter group and itsrepresentations”, Algebra Number Theory (2013), 595–671[HoLe1] R.B. Howlett, G. Lehrer, “Induced cuspidal representations and generalised Hecke rings”,Invent. Math. (1980), 37–64[HoLe2] R.B. Howlett, G. Lehrer, “Representations of generic algebras and finite groups of Lietype”, Trans. Amer. Math. Soc. (1983), 753–779[Hum] J.E. Humphreys, Reflection groups and Coxeter groups , Cambridge Studies in AdvancedMathematics , Cambridge University Press, 1990[Iwa1] N. Iwahori, “On the structure of a Hecke ring of a Chevalley group over a finite field”, J.Fac. Sci. Univ. Tokyo Sect. I (1964), 215–236[Iwa2] N. Iwahori, “Generalized Tits system (Bruhat decompostition) on p -adic semisimple groups”,pp. 71–83 in: Algebraic groups and discontinuous subgroups , Proc. Sympos. Pure Math. ,American Mathematical Society, 1966[IwMa] N. Iwahori, H. Matsumoto. “On some Bruhat decomposition and the structure of the Heckerings of the p -adic Chevalley groups”, Inst. Hautes ´Etudes Sci. Publ. Math (1965), 5–48[Jon] V.R.F. Jones, “Hecke algebra representations of braid groups and link polynomials”, Ann.Math. (1987), 335–388[Kat1] S.-I. Kato, “Irreducibility of principal series representations for Hecke algebras of affinetype”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. (1981), 929–943[Kat2] S.-I. Kato, “A realization of irreducible representations of affine Weyl groups”, Indag. Math. (1983), 193–201[KaLu1] D. Kazhdan, G. Lusztig, “Representations of Coxeter groups and Hecke algebras”, Invent.Math. (1979), 165–184[KaLu2] D. Kazhdan, G. Lusztig, “Proof of the Deligne–Langlands conjecture for Hecke algebras”,Invent. Math. (1987), 153–215[Kir] A. Kirillov, “Lectures on affine Hecke algebras and Macdonald’s conjectures”, Bull. Amer.Math. Soc. (1997), 251–292[Kna] A.W. Knapp, Representation theory of semisimple groups. An overview based on examples ,Princeton Mathematical Series , Princeton University Press, 1986[KrRa] C. Kriloff, A. Ram, “Representations of graded Hecke algebras”, Representation Theory (2002), 31–69[Lus1] G. Lusztig, “Intersection cohomology complexes on a reductive group”, Invent. Math. (1984), 205–272[Lus2] G. Lusztig “Cuspidal local systems and graded Hecke algebras”, Publ. Math. Inst. Hautes´Etudes Sci. (1988), 145–202 FFINE HECKE ALGEBRAS AND THEIR REPRESENTATIONS 75 [Lus3] G. Lusztig, “Affine Hecke algebras and their graded version”, J. Amer. Math. Soc (1989),599–635[Lus4] G. Lusztig, “Cuspidal local systems and graded Hecke algebras. II”, pp. 216–275 in:
Repre-sentations of groups , Canadian Mathematical Society Conference Proceedings , 1995[Lus5] G. Lusztig, “Classification of unipotent representations of simple p -adic groups” Int. Math.Res. Notices (1995), 517–589[Lus6] G. Lusztig, “Cuspidal local systems and graded Hecke algebras. III”, Represent. Theory (2002), 202-242[Lus7] G. Lusztig, Hecke algebras with unequal parameters , CRM Monograph Series , AmericanMathematical Society, 2003[Mac] I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials , Cambridge Tracts inMathematics , Cambridge University Press, 2003[Mat1] H. Matsumoto, “G´en´erateurs et relations des groupes de Weyl g´en´eralis´es”, C.R. Acad. Sci.Paris (1964), 3419–3422[Mat2] H. Matsumoto,
Analyse harmonique dans les syst`emes de Tits bornologiques de type affine ,Lecture Notes in Mathematics , Springer-Verlag, 1977[MiSt] M. Miyauchi, S. Stevens, “Semisimple types for p -adic classical groups”, Math. Ann. (2014), 257–288[Mor] L. Morris, “Tamely ramified intertwining algebras”, Invent. Math. (1993), 1–54[Opd1] E.M. Opdam, “Harmonic analysis for certain representations of graded Hecke algebras”,Acta Math. (1995), 75–121[Opd2] E.M. Opdam, “On the spectral decomposition of affine Hecke algebras”, J. Inst. Math.Jussieu (2004), 531–648[Opd3] E.M. Opdam, “Spectral transfer morphisms for unipotent affine Hecke algebras”, J. Inst.Math. Jussieu (2004), 531–648[Opd4] E.M. Opdam, “Spectral correspondences for affine Hecke algebras”, Advances Math. 286(2016), 912–957[OpSo1] E.M. Opdam, M. Solleveld, “Homological algebra for affine Hecke algebras”, Adv. Math. (2009), 1549–1601[OpSo2] E.M. Opdam, M. Solleveld, “Discrete series characters for affine Hecke algebras and theirformal dimensions”, Acta Math. (2010), 105–187[RaRa] A. Ram, J. Rammage, “Affine Hecke algebras, cyclotomic Hecke algebras and Cliffordtheory”, pp. 428–466 in: A tribute to C.S. Seshadri (Chennai 2002) , Trends in Mathematics,Birkh¨auser, 2003[Ree1] M. Reeder, “Nonstandard intertwining operators and the structure of unramified principalseries representations”, Forum Math. (1997), 457–516[Ree2] M. Reeder, “Isogenies of Hecke algebras and a Langlands correspondence for ramified prin-cipal series representations”, Representation Theory (2002), 101–126[Ren] D. Renard, Repr´esentations des groupes r´eductifs p-adiques , Cours sp´ecialis´es , Soci´et´eMath´ematique de France, 2010[Roc] A. Roche, “Types and Hecke algebras for principal series representations of split reductive p -adic groups”, Ann. Sci. ´Ecole Norm. Sup. (1998), 361–413[Sil] A.J. Silberger, Introduction to harmonic analysis on reductive p-adic groups , MathematicalNotes , Princeton University Press, 1979[Slo1] K. Slooten, A combinatorial generalization of the Springer correspondence for classical type ,PhD Thesis, Universiteit van Amsterdam, 2003[Slo2] K. Slooten, “Generalized Springer correspondence and Green functions for type B/C gradedHecke algebras”, Advances in Math. (2006), 34–108[Sol1] M. Solleveld,
Periodic cyclic homology of affine Hecke algebras , PhD Thesis, Universiteit vanAmsterdam, 2007[Sol2] M. Solleveld, “Homology of graded Hecke algebras”, J. Algebra (2010), 1622–1648[Sol3] M. Solleveld, “Parabolically induced representations of graded Hecke algebras”, Algebras andRepresentation Theory (2012), 233–271[Sol4] M. Solleveld, “On the classification of irreducible representations of affine Hecke algebraswith unequal parameters”, Representation Theory (2012), 1–87[Sol5] M. Solleveld “Hochschild homology of affine Hecke algebras”, J. Algebra (2013), 1–35 [Sol6] M. Solleveld, “Topological K-theory of affine Hecke algebras”, Ann. K-theory (2018),395–460[Sol7] M. Solleveld, “On completions of Hecke algebras”, pp. 207–262 in: Representations of Re-ductive p-adic Groups, A.-M. Aubert, M. Mishra, A. Roche, S. Spallone (eds.) , Progress inMathematics , Birkh¨auser, 2019[Sol8] M. Solleveld, “Endomorphism algebras and Hecke algebras for reductive p -adic groups”,arXiv:2005.07899, 2020[Spr] T.A. Springer, Linear algebraic groups 2nd ed. , Progress in Mathematics , Birkh¨auser, 1998[Wal] J.-L. Waldspurger, “La formule de Plancherel pour les groupes p -adiques (d’apr`es Harish-Chandra)”, J. Inst. Math. Jussieu (2003), 235–333[Yan] R. Yan, “Isomorphisms between affine Hecke algebras of type (cid:102) A ”, J. Algebra (2010),984–999[Zel] A.V. Zelevinsky, “Induced representations of reductive p -adic groups II. On irreducible repre-sentations of GL(n)”, Ann. Sci. ´Ecole Norm. Sup. (4)13.2