Higher Orbit Integrals, Cyclic Cocyles, and K-theory of Reduced Group C*-algebra
aa r X i v : . [ m a t h . K T ] N ov Higher Orbit Integrals, Cyclic Cocyles, and K-theory ofReduced Group C ∗ -algebra Yanli Song ∗ , Xiang Tang † Abstract
Let G be a connected real reductive group. Orbit integrals define traces on the groupalgebra of G . We introduce a construction of higher orbit integrals in the direction of highercyclic cocycles on the Harish-Chandra Schwartz algebra of G . We analyze these higher orbitintegrals via Fourier transform by expressing them as integrals on the tempered dual of G .We obtain explicit formulas for the pairing between the higher orbit integrals and the K -theory of the reduced group C ∗ -algebra, and discuss their applications to representationtheory and K -theory. Contents Φ x b G temp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 Proof of Theorem 4.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.6 Proof of Theorem 4.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 ∗ Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, MO, 63130, U.S.A.,[email protected]. † Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, MO, 63130, U.S.A.,[email protected]. Higher Index Pairing 24 K ( C ∗ r ( G )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Regular case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.4 Singular case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 A Integration of Schwartz functions 30B Characters of representations of G B.1 Discrete series representation of G . . . . . . . . . . . . . . . . . . . . . . . . . 33B.2 Discrete series representations of M . . . . . . . . . . . . . . . . . . . . . . . . 34B.3 Induced representations of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 C Description of K ( C ∗ r ( G )) C.1 Generalized Schmid identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35C.2 Essential representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
D Orbital integrals 38
D.1 Definition of orbital integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38D.2 The formula for orbital integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Let G be a connected real reductive group, and H a Cartan subgroup of G . Let f be acompactly supported smooth function on G . For x ∈ H reg , the integrals Λ Hf ( x ) := Z G/Z G ( x ) f ( gxg − ) d G/H ˙ g are important tools in representation theory with deep connections to number theory. Harish-Chandra showed the above integrals extend to all f in the Harish-Chandra Schwartz algebra S ( G ) , and obtained his famous Plancherel formula [6, 7, 8].In this paper, we aim to study the noncommutative geometry of the above integral andits generalizations. Let W ( H, G ) be the subgroup of the Weyl group of G consisting ofelements fixing H . Following Harish-Chandra, we define the orbit integral associated to H to be F H : S ( G ) → C ∞ ( H reg ) − W ( H,G ) , F Hf ( x ) := ǫ H ( x ) ∆ GH ( x ) Z G/Z G ( x ) f ( gxg − ) d G/H ˙ g, where C ∞ ( H reg ) − W ( H,G ) is the space of anti-symmetric functions with respect to the Weylgroup W ( H, G ) action on H , ǫ H ( h ) is a sign function on H , and ∆ GH is the Weyl denominatorfor H . Our starting point is the following property that for a given h ∈ H reg , the linearfunctional on S ( G ) , F H ( h ) : f → F Hf ( h ) , is a trace on S ( G ) , c.f. [11]. In cyclic cohomology, traces are special examples of cyclic cocy-cles on an algebra. In noncommutative geometry, there is a fundamental pairing betweenperiodic cyclic cohomology and K -theory of an algebra. The pairing between the orbit inte-grals F H ( h ) and K ( S ( G )) behaves differently between the cases when G is of equal rank andnon-equal rank. More explicitly, we will show in this article that when G has equal rank, F H defines an isomorphism as abelian groups from the K -theory of S ( G ) to the represen-tation ring of K , a maximal compact subgroup of G . Nevertheless, when G has non-equalrank, F H vanishes on K -theory of S ( G ) completely, c.f. [11]. Furthermore, many numerical nvariants for G -equivariant Dirac operators in literature, e.g . [1, 5, 11, 14, 22] etc, vanishwhen G has non-equal rank. Our main goal in this article is to introduce generalizationsof orbit integrals in the sense of higher cyclic cocycles on S ( G ) which will treat equal andnon-equal rank groups in a uniform way and give new interesting numerical invariants for G -equivariant Dirac operators. We remark that orbit integrals and cyclic homology of S ( G ) were well studied in literature, e.g. [2, 15, 16, 17, 18, 24]. Our approach here differs fromprior works in its emphasis on explicit cocycles.To understand the non-equal rank case better, we start with the example of the abeliangroup G = R , which turns out to be very instructive. Here S ( R ) is the usual algebra ofSchwartz functions on R with convolution product, and it carries a nontrivial degree onecyclic cohomology. Indeed we can define a cyclic cocycle ϕ on S ( R ) as follows, (c.f. [16,Prop. 1.4]: ϕ ( f , f ) = Z R xf (− x ) f ( x ) dx. (1.1)Under the Fourier transform, the convolution algebra S ( R ) is transformed into the Schwartzfunctions with pointwise multiplication, and the cocycle ϕ is transformed into somethingmore familiar: ^ ϕ (^ f , ^ f ) = Z b R ^ f d ^ f . (1.2)It follows from the Hochschild-Kostant-Rosenberg theorem that ^ ϕ generates the degree onecyclic cohomology of S ( b R ) , and accordingly ϕ generates the degree one cyclic cohomologyof S ( R ) .We notice that it is crucial to have the function x in Equation (1.1) to have the integralof ^ f d ^ f on S ( b R ) . And our key discovery is a natural generalization of the function x on ageneral connected real reductive group G . Let P = MAN be a parabolic subgroup of G . TheIwasawa decomposition G = KMAN writes an element g ∈ G as g = κ ( g ) µ ( g ) e H ( g ) n ∈ KMAN = G. Let dim ( A ) = n . And the function H = ( H , . . . , H n ) : G → a provides us the right ingredi-ent to generalize the cocycle ϕ in Equation (1.1). We introduce a generalization Φ P for orbitintegrals in Definition 3.3. For f , ..., f n ∈ S ( G ) and x ∈ M , Φ P,x is defined by the followingintegral, Φ P,x ( f , f , . . . , f n ):= Z h ∈ M/Z M ( x ) Z KN Z G × n X τ ∈ S n sgn ( τ ) · H τ ( ) ( g ...g n k ) H τ ( ) ( g ...g n k ) . . . H τ ( n ) ( g n k ) f (cid:0) khxh − nk − ( g . . . g n ) − (cid:1) f ( g ) . . . f n ( g n ) dg · · · dg n dkdndh, where Z M ( x ) is the centralizer of x in M . Though the function H is not a group cocycle on G , we show in Lemma 3.1 that it satisfies a kind of twisted group cocycle property, whichleads us to the following theorem in Section 3.1. Theorem I. (Theorem 3.5) For a maximal parabolic subgroup P ◦ = M ◦ A ◦ N ◦ and x ∈ M ◦ , thecochain Φ P ◦ ,x is a continuous cyclic cocycle of degree n on S ( G ) . Modeling on the above example of R , e.g. Equation (1.2), we analyzes the higher orbitintegral Φ P by computing its Fourier transform. Using Harish-Chandra’s theory of orbit in-tegrals and character formulas for parabolic induced representations, we introduce in Defi-nition 4.14 a cyclic cocycle b Φ x defined as an integral on b G . The following theorem in Section4 establishes a full generalization of Equation (1.2) to connected real reductive groups. heorem II. (Theorem 4.15 and 4.18) Let T be a compact Cartan subgroup of M ◦ . For any t ∈ T reg ,and f , . . . f n ∈ S ( G ) , the following identity holds, Φ P ◦ ,e ( f , . . . , f n ) = (− ) n b Φ e ( b f , . . . , b f n ) ,∆ M ◦ T ( t ) · Φ P ◦ ,t ( f , . . . , f n ) = (− ) n b Φ t ( b f , . . . , b f n ) , where b f , ..., b f n are the Fourier transforms of f , · · · , f n ∈ S ( G ) , and ∆ M ◦ T ( t ) is the Weyl denomi-nator of T . As an application of our study, we compute the pairing between the K -theory of S ( G ) and Φ P . Vincent Lafforgue showed in [13] that Harish-Chandra’s Schwartz algebra S ( G ) is a subalgebra of C ∗ r ( G ) , stable under holomorphic functional calculus. Therefore, the K -theory of S ( G ) is isomorphic to the K -theory of the reduced group C ∗ -algebra C ∗ r ( G ) . Thestructure of C ∗ r ( G ) is studied by [3, 23]. As a corollary, we are able to explicitly identify [4]a set of generators of the K -theory of C ∗ r ( G ) as a free abelian group, c.f. Theorem C.3. Withwave packet, we construct a representative [ Q λ ] ∈ K ( C ∗ r ( G )) for each generator in TheoremC.3. Applying Harish-Chandra’s theory of orbit integrals, we compute explicitly in the fol-lowing theorem the index pairing between [ Q λ ] and Φ P . Theorem III. (Theorem 5.4) The index pairing between cyclic cohomology and K -theory HP even (cid:0) S ( G ) (cid:1) ⊗ K (cid:0) S ( G ) (cid:1) → C is given by the following formulas: • We have h Φ e , [ Q λ ] i = (− ) dim ( A P ) | W M ◦ ∩ K | · X w ∈ W K m (cid:0) σ M ◦ ( w · λ ) (cid:1) , where σ M ◦ ( w · λ ) is the discrete series representation with Harish-Chandra parameter w · λ ,and m (cid:0) σ M ◦ ( w · λ ) (cid:1) is its Plancherel measure; • For any t ∈ T reg , we have that h Φ t , [ Q λ ] i = (− ) dim ( A P ) P w ∈ W K (− ) w e w · λ ( t ) ∆ M ◦ T ( t ) . We refer the readers to Theorem 5.4 for the notations involved the above formulas. Forthe case of equal rank, the first formula was obtained in [5] in which Connes-Moscovicciused the L -index on homogeneous spaces to detect the Plancherel measure of discreteseries representations. It is interesting to point out, c.f. Remark 3.7, that the higher or-bit integrals Φ P ◦ ,x actually extend to a family of Banach subalgebras of C ∗ r ( G ) introducedby Lafforgue, [13, Definition 4.1.1]. However, we have chosen to work with the Harish-Chandra Schwartz algebra S ( G ) as our proofs rely crucially on Harish-Chandra’s theory oforbit integrals and character formulas.Note that the higher orbit integrals Φ P,x reduce to the classical ones when G is equalrank. Nevertheless, our main results, Theorem II. and III. for higher orbit integrals, are alsonew in the equal rank case. For example, as a corollary to Theorem III., in Corollary 5.5, weare able to detect the character information of limit of discrete series representations usingthe higher orbit integrals. This allows us to identify the contribution of limit of discreteseries representations in the K -theory of C ∗ r ( G ) without using geometry of the homogeneousspace G/K , e.g. the Connes-Kasparov index map. As an application, our computation ofthe index pairing in Theorem III. suggests a natural isomorphism F T , Definition 5.7 andCorollary 5.8, F T : K ( C ∗ r ( G )) → Rep ( K ) . n [4], we will prove that F T is the inverse of the Connes-Kasparov index map,Ind : Rep ( K ) → K ( C ∗ r ( G )) . Our development of higher orbit integrals raises many intriguing questions. Let us listtwo of them here. • Given a Dirac operator D on G/K , the Connes-Kasparov index map gives an elementInd ( D ) in K ( C ∗ r ( G )) . In this article, Theorem III. and its corollaries, we study the rep-resentation theory information of the index pairing h [ Φ P ] , [ Ind ( D )] i . Is there a topological formula for the above pairing, generalizing the Connes-Moscovici L -index theorem [5]? • In this article, motivated by the applications in K -theory, we introduce Φ P,x as a cycliccocycle on S ( G ) . Actually, the construction of Φ P,x can be generalized to construct alarger class of Hochschild cocycles for S ( G ) . For example, for a general (not necessarilymaximal) parabolic subgroup P , there are corresponding versions of Theorem I. andII., which are related to more general differentiable currents on b G . How are thesedifferentiable currents related to the Harish-Chandra’s Plancherel theory?The article is organized as follows. In Section 2, we review some basics about represen-tation theory of real reductive Lie groups, Harish-Chandra’s Schwartz algebra, and cyclictheory. We introduce the higher orbit integral Φ P in Section 3 and prove Theorem I. TheFourier transform of the higher orbit integral is studied in Section 4 with the proof of Theo-rem II. And in Section 5, we compute the pairing between the higher orbit integrals Φ P and K ( C ∗ r ( G )) , proving Theorem III., and its corollaries. For the convenience of readers, we haveincluded in the appendix some background review about related topics in representationtheory and K ( C ∗ r ( G )) . Acknowledgments:
We would like to thank Nigel Higson, Peter Hochs, Markus Pflaum,Hessel Posthuma and Hang Wang for inspiring discussions. Our research are partially sup-ported by National Science Foundation. We would like to thank Shanghai Center of Math-ematical Sciences for hosting our visits, where parts of this work were completed.
In this article, we shall not attempt to strive for the utmost generality in the class of groupswe shall consider. Instead we shall aim for (relative) simplicity.
Let G ⊆ GL ( n, R ) be a self-adjoint group which is also the group of real points of a con-nected algebraic group defined over R . For brevity, we shall simply say that G is a real re-ductive group . In this case, the Cartan involution on the Lie algebra g is given by θ ( X ) = − X T ,where X T denotes the transpose matrix of X .Let K = G ∩ O ( n ) , which is a maximal compact subgroup of G . Let k be the Lie algebraof K . We have a θ -stable Cartan decomposition g = k ⊕ p . Let A be a maximal abelian subgroup of positive definite matrices in G . And use a todenote the associated Lie algebra of A . Fix a positive chamber a + ⊆ a . Denote the associated evi subgroup by L . L can be written canonically as a Cartesian product of two groups M and A , i.e. L = M × A . Let N be the corresponding unipotent subgroup, and P be theassociated parabolic subgroup. We have ( M × A ) ⋉ N = MAN.
Let T be a compact Cartan subgroup of M . Then H = TA gives a Cartan subgroup of G . Note that there is a one to one correspondence between the Cartan subgroup H and thecorresponding parabolic subgroup P . We say that H is the associated Cartan subgroup forthe parabolic subgroup P and vice versa.Let h be the Lie algebra of H . Denote by R (h , g) the set of roots. We can decompose R (h , g) into a union of compact and non-compact roots: R (h , g) = R c (h , g) ⊔ R n (h , g) , where R c (h , g) = R (h , k) and R n (h , g) = R (h , p) . For a non-compact imaginary root β in R (h , g) , we can apply the Cayley transform to h and obtain a more non-compact Cartansubalgebra, together with the corresponding new parabolic subgroup. In this case, we saythat the new parabolic subgroup is more noncompact than the original one. This defines apartial order on the set of all parabolic subgroups of G . Definition 2.1.
If the associated Cartan subgroup of a parabolic subgroup is the most com-pact (i.e the compact part has the maximal dimension), then we call it the maximal parabolicsubgroup , and denote it by P ◦ . Accordingly, we write P ◦ = M ◦ A ◦ N ◦ and the Cartan sub-group H ◦ = T ◦ A ◦ . Let π be a spherical principal series representation of G , that is, the unitary representationinduced from the trivial representation of a minimal parabolic subgroup. Let v be a unit K -fixed vector in the representation space of π , known as spherical vector. Let Ξ be thematrix coefficient of v , i.e. for all g ∈ G , Ξ ( g ) = h v, gv i π . The inner product on g defines a G-invariant Riemannian metric on G/K . For g ∈ G , let k g k be the Riemannian distance from eK to gK in G/K . For every m ≥
0, X, Y ∈ U (g) , and f ∈ C ∞ ( G ) , set ν X,Y,m ( f ) := sup g ∈ G (cid:10) ( + k g k )) m Ξ ( g ) − (cid:12)(cid:12) L ( X ) R ( Y ) f ( g ) (cid:12)(cid:12) (cid:11) , where L and R denote the left and right regular representations, respectively. Definition 2.2.
The Harish-Chandra Schwartz space S ( G ) is the space of f ∈ C ∞ ( G ) suchthat for all m ≥ and X, Y ∈ U (g) , ν X,Y,m ( f ) < ∞ .The space S ( G ) is a Fr´echet space in the semi- norms ν X,Y,m . It is closed under convolu-tion, which is a continuous operation on this space. Moreover, if G has a discrete series, thenall K -finite matrix coefficients of discrete series representations lie in S ( G ) . It is proved in[13] that S ( G ) is a ∗ -subalgebra of the reduced group C ∗ -algebra C ∗ r ( G ) that is closed underholomorphic functional calculus. .3 Cyclic cohomology Definition 2.3.
Let A be an algebra over C . Define the space of Hochschild cochains ofdegree k of A by C k ( A ) : = Hom C (cid:0) A ⊗ ( k + ) , C (cid:1) of all bounded k + -linear functionals on A . Define the Hochschild codifferential b : C k ( A ) → C k + ( A ) by bΦ ( a ⊗ · · · ⊗ a k + )= k X i = (− ) i Φ ( a ⊗ · · · ⊗ a i a i + ⊗ · · · ⊗ a k + ) + (− ) k + Φ ( a k + a ⊗ a ⊗ · · · ⊗ a k ) . The Hochschild cohomology is the cohomology of the complex ( C ∗ ( A ) , b ) . Definition 2.4.
We call a k -cochain Φ ∈ C k ( A ) cyclic if for all a , . . . , a k ∈ A it holds that Φ ( a k , a , . . . , a k − ) = (− ) k Φ ( a , a , . . . , a k ) . The subspace C kλ of cyclic cochains is closed under the Hochschild codifferential. Andthe cyclic cohomology HC ∗ ( A ) is defined by the cohomology of the subcomplex of cycliccochains.Let R = ( R i,j ) , i, j =
1, . . . , n be an idempotent in M n ( A ) . The following formula ! n X i , ··· ,i = Φ ( R i i , R i i , ..., R i i ) defines a natural pairing between [ Φ ] ∈ HP even ( A ) and K ( A ) , i.e. h · , · i : HP even ( A ) ⊗ K ( A ) → C . In this section, we construct higher orbit integrals as cyclic cocycles on S ( G ) for a maximalparabolic subgroup P ◦ of G . Let P = MAN be a parabolic subgroup and denote n = dim A . By the Iwasawa decompo-sition, we have that G = KMAN.
We define a map H = ( H , . . . , H n ) : G → a by the following decomposition g = κ ( g ) µ ( g ) e H ( g ) n ∈ KMAN = G. The above decomposition is not unique since K ∩ M = ∅ . Nevertheless the map H is well-defined. Lemma 3.1.
For any g , g ∈ G , the function H i ( g κ ( g )) does not depend on the choice of κ ( g ) .Moreover, the following identity holds H i ( g ) + H i ( g κ ( g )) = H i ( g g ) . roof. We write g = k m a n , g = k m a n . Recall that the group MA normalizes N and M commutes with A . Thus, H i ( g k ) = H i ( k m a n k ) = H i ( k km a n ′ ) = H i ( a ) = H i ( g ) . for any k ∈ K ∩ M . It follows that H i ( g κ ( g )) is well-defined. Next, H i ( g g ) = H i ( a n k a ) = H i ( a n k ) + H i ( a ) . The lemma follows from the following identities H i ( g κ ( g )) = H i ( a n k ) , H i ( g ) = H i ( a ) . Let S n be the permutation group of n elements. For any τ ∈ S n , let sgn ( τ ) = ± de-pending on the parity of τ . Definition 3.2.
We define a function C ∈ C ∞ (cid:0) K × G × n (cid:1) by C ( k, g , . . . , g n ) := X τ ∈ S n sgn ( τ ) · H τ ( ) ( g k ) H τ ( ) ( g k ) . . . H τ ( n ) ( g n k ) . Definition 3.3.
For any f , . . . , f n ∈ S ( G ) and x ∈ M , we define a Hochschild cochain on S ( G ) by the following formula Φ P,x ( f , f , . . . , f n ) : = Z h ∈ M/Z M ( x ) Z KN Z G × n C ( k, g g . . . g n , . . . , g n − g n , g n ) f (cid:0) khxh − nk − ( g . . . g n ) − (cid:1) f ( g ) . . . f n ( g n ) dg · · · dg n dkdndh, (3.1)where Z M ( x ) is the centralizer of x in M .We prove in Theorem A.5 that the above integral (3.1) is convergent for x ∈ M . A similarestimate leads us to the following property. Proposition 3.4.
For all x ∈ M , the integral Φ P,x defines a (continuous) Hochschild cochain onthe Schwarz algebra S ( G ) . In this paper, By abusing the notations, we write Φ x for Φ P ◦ ,x . Furthermore, for sim-plicity, we omit the respective measures dg , · · · , dg n , dk, dn, dh , in the integral (3.1) for Φ x . Theorem 3.5.
For a maximal parabolic subgroup P ◦ = M ◦ A ◦ N ◦ and x ∈ M ◦ , the cochain Φ x is acyclic cocycle and defines an element [ Φ x ] ∈ HC n ( S ( G )) . Remark 3.6.
In fact, our proof shows that Φ P,x are cyclic cocycles for all parabolic subgroup P and x ∈ M . For the purpose of this paper, we will focus on the case when P = P ◦ ismaximal, as Φ P ◦ ,x , x ∈ M , generate the whole cyclic cohomology of S ( G ) , c.f. Remark 5.9. Remark 3.7.
We notice that our proofs in Sec. 3.2 and 3.3 also work for the algebra S t ( G ) (for sufficiently large t ) introduced in Definition A.3. And we can conclude from TheoremA.5 that Φ x defines a continuous cyclic cocycle on S t ( G ) ⊃ S ( G ) for a sufficiently large t forevery x ∈ M .The proof of Theorem 3.5 occupies the left of this section. .2 Cocycle condition In this subsection, we prove that the cochain Φ x introduced in Definition 3.3 is a Hochschildcocycle.We have the following expression for the codifferential of Φ x , i.e. bΦ x ( f , f , . . . , f n , f n + )= n X i = (− ) i Φ x (cid:0) f , . . . , f i ∗ f i + , . . . , f n + (cid:1) + (− ) n + Φ x (cid:0) f n + ∗ f , f , . . . , f n (cid:1) . (3.2)Here f i ∗ f i + is the convolution product given by f i ∗ f i + ( h ) = Z G f i ( g ) f i + ( g − h ) dg. When i = , the first term in the expression of bΦ x (See Equation (3.2)) is computed by thefollowing integral, Φ x (cid:0) f ∗ f , f , . . . , f n + (cid:1) = Z M/Z M ( x ) Z KN Z G Z G × n C ( k, g g . . . g n , . . . , g n − g n , g n ) · f ( g ) f (cid:0) g − · khxh − nk − ( g . . . g n ) − (cid:1) · f ( g ) . . . f n + ( g n ) . (3.3)By changing variables, t = g − · khxh − nk − ( g . . . g n ) − , t j = g j − , j =
2, . . . n + we get g = khxh − nk − ( t . . . t n + ) − . We can rewrite (3.3) into Φ x (cid:0) f ∗ f , f , . . . , f n + (cid:1) = Z M/Z M ( x ) Z KN Z G × ( n + ) C ( k, t t . . . t n + , . . . , t n t n + , t n + ) f (cid:0) khxh − nk − ( t . . . t n + ) − (cid:1) f ( t ) · f ( t ) . . . f n + ( t n + ) . (3.4)For ≤ i ≤ n , we have Φ x (cid:0) f , . . . , f i ∗ f i + , . . . , f n + (cid:1) = Z M/Z M ( x ) Z KN Z G Z G × n C ( k, g g . . . g n , . . . , g n − g n , g n ) f (cid:0) khxh − nk − ( g . . . g n ) − (cid:1) f ( g ) . . . f i − ( g i − ) (cid:0) f i ( g ) f i + ( g − g i ) (cid:1) · f i + ( g i + ) . . . f n + ( g n ) . (3.5)Let t j = g j for j =
1, . . . i − , and t i = g, t i + = g − g i , t j = g j − , j = i +
2, . . . , n + We rewrite (3.5) as Φ x (cid:0) f , . . . , f i ∗ f i + , . . . , f n + (cid:1) = Z M/Z M ( x ) Z KN Z G × ( n + ) C ( k, t t . . . t n + , . . . , ( t i + . . . t n + ) ^ , . . . , t n + ) · f (cid:0) khxh − nk − ( t . . . t n + ) − (cid:1) f ( t ) . . . f n + ( t n + ) , (3.6) here ( t i + . . . t n + ) ^ means that the term is omitted in the expression.Now we look at the last term in the expression of bΦ x , (c.f. Equation (3.2)), Φ x (cid:0) f n + ∗ f , f , . . . , f n (cid:1) = Z M/Z M ( x ) Z KN Z G Z G × n C ( k, g g . . . g n , . . . , g n − g n , g n ) · f n + ( g ) f (cid:0) g − · khmh − nk − ( g . . . g n ) − (cid:1) · f ( g ) . . . f n ( g n ) . (3.7)As before, we denote by t j = g j , j =
1, . . . , n, and t n + = g . We can rewrite Equation (3.7) as Φ x (cid:0) f n + ∗ f , f , . . . , f n (cid:1) = Z M/Z M ( x ) Z KN Z G × ( n + ) C ( k, t t . . . t n , . . . , t n − t n , t n ) · f (cid:0) t − + · khxh − nk − ( t . . . t n ) − (cid:1) · f ( t ) . . . f n + ( t n + ) . (3.8) Lemma 3.8. Φ x (cid:0) f n + ∗ f , f , . . . , f n (cid:1) = Z M/Z M ( x ) Z KN Z G × ( n + ) C ( κ ( t n + k ) , t t . . . t n , . . . , t n − t n , t n ) · f (cid:0) khxh − nk − ( t . . . t n + ) − (cid:1) · f ( t ) . . . f n + ( t n + ) . (3.9) Proof.
Using G = KMAN , we decompose t − + · k = k m a n ∈ KMAN.
It follows that k = t n + k m a n and k = κ ( t n + k ) . We see Φ x (cid:0) f n + ∗ f , f , . . . , f n (cid:1) = Z M/Z M ( x ) Z KN Z G × ( n + ) C ( k, t t . . . t n , . . . , t n − t n , t n ) f (cid:0) k m a n hxh − nn − a − m − k − t − + ( t . . . t n ) − (cid:1) · f ( t ) . . . f n + ( t n + ) . Since m a ∈ MA normalizes the nilpotent group N , we have the following identity, f (cid:0) k m a n hxh − nn − a − m − k − t − + ( t . . . t n ) − (cid:1) = f (cid:0) k m hx ( m h ) − ˜ n nn ′ − k − ( t . . . t n + ) − (cid:1) , where ˜ n , n ′ are defined by n and m , a , h, and x . Therefore, renaming ˜ n nn ′ − by n ,we have Φ P,x (cid:0) f n + ∗ f , f , . . . , f n (cid:1) = Z M/Z M ( x ) Z KN Z G × ( n + ) C ( κ ( t n + k ) , t t . . . t n , . . . , t n − t n , t n ) f (cid:0) k hxh − nk − ( t . . . t n + ) − (cid:1) · f ( t ) . . . f n + ( t n + ) . This completes the proof.Combining (3.4), (3.6) and (3.9), we have reached the following lemma. emma 3.9. The codifferential bΦ GP,x ( f , f , . . . , f n , f n + )= Z M/Z M ( x ) Z KN Z G × ( n + ) ˜ C ( k, t , . . . t n + ) · f (cid:0) k hxh − nk − ( t . . . t n + ) − (cid:1) · f ( t ) . . . f n + ( t n + ) , where ˜ C ∈ C ∞ (cid:0) K × G × n (cid:1) is given by ˜ C ( k, t , . . . t n + ) = n X i = (− ) i C ( k, t t . . . t n + , . . . , ( t i + . . . t n + ) ^ , . . . , t n + )+(− ) n + C ( κ ( t n + k ) , t t . . . t n , . . . , t n − t n , t n ) . Lemma 3.10. ˜ C ( k, t , . . . t n + ) = Proof.
To begin with, we notice the following expression, C (cid:0) κ ( t n + k ) , t t . . . t n , . . . , t n − t n , t n (cid:1) = X τ ∈ S n sgn ( τ ) · H τ ( ) (cid:0) t . . . t n κ ( t n + k ) (cid:1) H τ ( ) (cid:0) t . . . t n κ ( t n + k ) (cid:1) . . . H τ ( n ) (cid:0) t n κ ( t n + k ) (cid:1) . By Lemma 3.1, we have H τ ( i ) (cid:0) t i . . . t n κ ( t n + k ) (cid:1) = H τ ( i ) (cid:0) t i . . . t n + k (cid:1) − H τ ( i ) ( t n + k ) . Using the above property of H τ ( i ) , we have C (cid:0) κ ( t n + k ) , t t . . . t n , . . . , t n − t n , t n (cid:1) = X τ ∈ S n sgn ( τ ) · (cid:16) H τ ( ) ( t . . . t n + k ) − H τ ( ) ( t n + k ) (cid:17) · (cid:16) H τ ( ) ( t . . . t n + k ) − H τ ( ) ( t n + k ) (cid:17) . . . (cid:16) H τ ( n ) ( t n t n + k ) − H τ ( n ) ( t n + k ) (cid:17) . By the symmetry of the permutation group S n , in the above summation, the terms cancontain at most one H τ ( i ) ( t n + k ) . Otherwise, it will be cancelled. Thus, C (cid:0) κ ( t n + k ) , t t . . . t n , . . . , t n − t n , t n (cid:1) = n X i = X τ ∈ S n sgn ( τ ) · H τ ( ) ( t . . . t n + k ) . . . (cid:16) − H τ ( i ) ( t n + k ) (cid:17) . . . H τ ( n ) ( t n t n + k )+ X τ ∈ S n sgn ( τ ) · H τ ( ) ( t . . . t n + k ) . . . H τ ( n ) ( t n t n + k ) . In the above expression, by changing the permutations (cid:0) τ ( ) , . . . , τ ( n ) (cid:1) → (cid:0) τ ( ) , . . . , τ ( i − ) , τ ( n ) , τ ( i ) , . . . , τ ( n − ) (cid:1) , we get X τ ∈ S n sgn ( τ ) · H τ ( ) ( t . . . t n + k ) . . . H τ ( i − ) ( t i − . . . t n + k ) (cid:16) − H τ ( i ) ( t n + k ) (cid:17) H τ ( i + ) ( t i + . . . t n + k ) . . . H τ ( n ) ( t n t n + k )=(− ) n − i X τ ∈ S n sgn ( τ ) · H τ ( ) ( t . . . t n + k ) . . . H τ ( i − ) ( t i − . . . t n k ) H τ ( n ) ( t n + k ) H τ ( i ) ( t i + . . . t n + k ) . . . H τ ( n − ) ( t n t n + k ) . utting all the above together, we have (− ) n + C (cid:0) κ ( t n + k ) , t t . . . t n , . . . , t n − t n , t n (cid:1) = n X i = (− ) i + · C ( k, t t . . . t n + , . . . , ( t i + . . . t n + ) ^ , . . . , t n + ) , and ˜ C ( k, t , . . . t n + ) = n X i = (− ) i C ( k, t t . . . t n + , . . . , ( t i + . . . t n + ) ^ , . . . , t n + )+(− ) n + C ( κ ( t n + k ) , t t . . . t n , . . . , t n − t n , t n ) = We conclude from Lemma 3.9 and Lemma 3.10 that Φ x is a Hochschild cocycle. In this subsection, we prove that the cocycle Φ x is cyclic. Recall Φ x ( f , . . . , f n , f ) = Z M/Z M ( x ) Z KN Z G × n C ( k, g g . . . g n , . . . , g n − g n , g n ) f (cid:0) khxh − nk − ( g . . . g n ) − (cid:1) f ( g ) . . . f n ( g n − ) f ( g n ) . By changing the variables, t = khxh − nk − ( g . . . g n ) − , t j = g j − , j =
2, . . . n − we have g n = ( t . . . t n ) − khxh − nk − , and g i . . . g n = ( t . . . t i ) − khxh − nk − . It follows that Φ x ( f , . . . , f n , f )= Z M/Z M ( x ) Z KN Z G × n C (cid:0) k, t − khxh − nk − , . . . , ( t . . . t n − ) − khxh − nk − , ( t . . . t n ) − khxh − nk − (cid:1) · f (cid:0) ( t . . . t n ) − khxh − nk − (cid:1) f ( t ) f ( t ) . . . f n ( t n )= X τ ∈ S n sgn ( τ ) · Z M/Z M ( x ) Z KN Z G × n H τ ( ) ( t − khxh − n ) . . . H τ ( n ) (( t . . . t n ) − khxh − n ) f (cid:0) ( t . . . t n ) − khxh − nk − (cid:1) f ( t ) f ( t ) . . . f n ( t n ) . We write ( t . . . t n ) − k = k m a n ∈ KMAN.
Then k = ( t . . . t n ) k m a n , (3.10)and f (cid:0) ( t . . . t n ) − khxh − nk − (cid:1) = f (cid:0) k m a n hxh − nn − a − m − k − ( t . . . t n ) − (cid:1) = f (cid:0) k h ′ xh ′ − n ′ k − ( t . . . t n ) − (cid:1) . hus, Φ x ( f , . . . , f n , f ) = X τ ∈ S n sgn ( τ ) · Z M/Z M ( x ) Z KN Z G × n H τ ( ) ( t − k ) . . . H τ ( n ) (( t . . . t n ) − k ) · f (cid:0) k hxh − nk − ( t . . . t n ) − (cid:1) f ( t ) f ( t ) . . . f n ( t n ) . By Lemma 3.1 and (3.10), for ≤ i ≤ n − , we have H τ ( i ) (cid:0) ( t . . . t i ) − k (cid:1) = − H τ ( i ) (cid:0) t . . . t i κ (( t . . . t i ) − k ) (cid:1) = − H τ ( i ) (cid:0) t . . . t i κ ( t i + . . . t n k ) (cid:1) = H τ ( i ) (cid:0) t i + . . . t n k (cid:1) − H τ ( i ) (cid:0) t . . . t n k ) (cid:1) and H τ ( n ) (cid:0) ( t . . . t n ) − k (cid:1) = − H τ ( n ) (cid:0) t . . . t n κ (( t . . . t n ) − k ) (cid:1) = − H τ ( n ) (cid:0) t . . . t n k (cid:1) . It follows that Φ x ( f , . . . , f n , f )= X τ ∈ S n sgn ( τ ) · Z M/Z M ( m ) Z KN Z G × n n − Y i = (cid:16) H τ ( i ) (cid:0) t i + . . . t n k (cid:1) − H τ ( i ) (cid:0) t . . . t n k ) (cid:1)(cid:17)(cid:0) − H τ ( n ) ( t . . . t n k ) (cid:1) · f (cid:0) k hxh − nk − ( t . . . t n ) − (cid:1) f ( t ) f ( t ) . . . f n ( t n ) . Note that X τ ∈ S n sgn ( τ ) · n − Y i = (cid:16) H τ ( i ) (cid:0) t i + . . . t n k (cid:1) − H τ ( i ) (cid:0) t . . . t n k ) (cid:1)(cid:17) · H τ ( n ) ( t . . . t n k )= X τ ∈ S n sgn ( τ ) · n − Y i = H τ ( i ) (cid:0) t i + . . . t n k (cid:1) · H τ ( n ) ( t . . . t n k ) . (3.11)In the above expression, by changing the permutation (cid:0) τ ( ) , . . . , τ ( n ) (cid:1) → (cid:0) τ ( ) , . . . , τ ( n ) , τ ( ) (cid:1) , we can simplify Equation (3.11) to the following one, X τ ∈ S n sgn ( τ ) · n − Y i = (cid:16) H τ ( i ) (cid:0) t i + . . . t n k (cid:1) − H τ ( i ) (cid:0) t . . . t n k ) (cid:1)(cid:17) · H τ ( n ) ( t . . . t n k )=(− ) n − · X τ ∈ S n sgn ( τ ) · n Y i = H τ ( i ) (cid:0) t i . . . t n k (cid:1) . Therefore, we have obtained the following identity, Φ x ( f , . . . , f n , f ) =(− ) n · X τ ∈ S n sgn ( τ ) · Z M/Z M ( x ) Z KN Z G × n n Y i = H τ ( i ) (cid:0) t i . . . t n k (cid:1) · f (cid:0) khxh − nk − ( t . . . t n ) − (cid:1) f ( t ) f ( t ) . . . f n ( t n )=(− ) n · Φ P,x ( f , . . . , f n ) . Hence, we conclude that Φ x is a cyclic cocycle, and have completed the proof of Theo-rem 3.5. The Fourier transform of Φ x In this section, we study the Fourier transform of the cyclic cocycle Φ x introduced in Sec-tion 3. For the convenience of readers, we start with recalling the basic knowledge aboutparabolic induction and the Plancherel formula in Section 4.1 and 4.2. A brief introduction to discrete series representations can be found in Appendix B. In thissection, we review the construction of parabolic induction. Let H be a θ -stable Cartan sub-group of G with Lie algebra h . Then h and H have the following decompositions, h = h k + h p , h k = h ∩ k , h p = h ∩ p , and H K = H ∩ K , H p = exp (h p ) . Let P be a parabolic subgroup of G with the split part H p ,that is P = M P H p N P = M P A P N P . Definition 4.1.
Let σ be a (limit of) discrete series representation of M P and ϕ a unitarycharacter of A P . The product σ ⊗ ϕ defines a unitary representation of the Levi subgroup L = M P A P . A basic representation of G is a representation by extending σ ⊗ ϕ to P triviallyacross N P then inducing to G : π σ,ϕ = Ind GP ( σ ⊗ ϕ ) . If σ is a discrete series then Ind GP ( σ ⊗ ϕ ) will be called a basic representation induced fromdiscrete series . This is also known as parabolic induction .The character of π σ,ϕ is given in Theorem B.5, Equation (B.3), and Corollary B.6. Notethat basic representations might not be irreducible. Knapp and Zuckerman complete theclassification of tempered representations by showing which basic representations are ir-reducible.Now consider a single parabolic subgroup P ⊆ G with associated Levi subgroup L , andform the group W ( A P , G ) = N K (a P ) /Z K (a P ) , where N K (a P ) and Z K (a P ) are the normalizer and centralizer of a P in K respectively. Thegroup W ( A P , G ) acts as outer automorphism of M P , and also the set of equivalence classesof representations of M P . For any discrete series representation σ of M P , we define W σ = (cid:8) w ∈ N K (a P ) : Ad ∗ w σ ∼ = σ (cid:9) /Z K (a P ) . Then the above Weyl group acts on the family of induced representations (cid:8)
Ind GP ( σ ⊗ ϕ ) (cid:9) ϕ ∈ b A P . Definition 4.2.
Let P and P be two parabolic subgroups of G with Levi factors L i = M P i A P i . Let σ and σ be two discrete series representations of M P i . We say that ( P , σ ) ∼ ( P , σ ) if there exists an element w in G that conjugates the Levi factor of P to the Levi factor of P , and conjugates σ to a representation unitarily equivalent to σ . In this case, there is aunitary G -equivariant isomorphismInd GP ( σ ⊗ ϕ ) ∼ = Ind GP ( σ ⊗ ( Ad ∗ w ϕ )) Knapp and Zuckerman prove that every tempered representation of G is basic, and every basic representationis tempered. hat covers the isomorphism Ad ∗ w : C ( b A P ) → C ( b A P ) . We denote by [ P, σ ] the equivalence class of ( P, σ ) , and P ( G ) the set of all equivalence classes.At last, we recall the functoriality of parabolic induction. Lemma 4.3. If S = M S A S N S is any parabolic subgroup of L , then the unipotent radical of SN P is N S N P , and the product Q = M Q A Q N Q = M S ( A S A P )( N S N P ) is a parabolic subgroup of G .Proof. See [20, Lemma 4.1.1].
Theorem 4.4 (Induction in stage) . Let η be a unitary representation (not necessarily a discreteseries representation) of M S . We decompose ϕ = ( ϕ , ϕ ) ∈ b A S × b A P . There is a canonical equivalence
Ind GP (cid:0) Ind M P S ( η ⊗ ϕ ) ⊗ ϕ (cid:1) ∼ = Ind GQ (cid:0) η ⊗ ( ϕ , ϕ ) (cid:1) . Proof.
See [12, P. 170].
Let G be a connected, linear, real reductive Lie group as before, and b G temp be the set ofequivalence classes of irreducible unitary tempered representations of G . For a Schwartzfunction f on G , its Fourier transform b f is defined by b f ( π ) = Z G f ( g ) π ( g ) dg, π ∈ b G temp . Thus, the Fourier transform assigns to f a family of operators on different Hilbert spaces π .The group A P , which consists entirely of positive definite matrices, is isomorphic to itsLie algebra via the exponential map. So A P carries the structure of a vector space, and wecan speak of its space of Schwartz functions in the ordinary sense of harmonic analysis. Thesame goes for the unitary (Pontryagin) dual b A P . By a tempered measure on A P we meana smooth measure for which integration extends to a continuous linear functional on theSchwartz space. Recall Harish-Chandra’s Plancherel formula for G , c.f. [8]. Theorem 4.5.
There is a unique smooth, tempered, W σ -invariant function m P,σ on the spaces b A P such that k f k ( G ) = X [ P,σ ] Z b A P (cid:13)(cid:13)(cid:13)b f ( π σ,ϕ ) (cid:13)(cid:13)(cid:13) m P,σ ( ϕ ) dϕ for every Schwartz function f ∈ S ( G ) . We call m P,σ ( ϕ ) the Plancherel density of the representation Ind GP ( σ ⊗ ϕ ) . As ϕ ∈ b A P varies, the Hilbert spaces π σ,ϕ = Ind GP ( σ ⊗ ϕ ) can be identified with one another as representations of K . Denote by Ind GP σ this commonHilbert space, and L ( Ind GP σ ) the space of K -finite Hilbert-Schmidt operators on Ind GP σ . Weshall discuss the adjoint to the Fourier transform. efinition 4.6. Let h be a Schwarz-class function from b A P into operators on Ind GP σ suchthat it is invariant under the W σ -action. That is h ∈ (cid:2) L ( b A P ) ⊗ L ( Ind GP σ ) (cid:3) W σ . The wave packet associated to h is the scalar function defined by the following formulaˇ h ( g ) = Z b A P Trace (cid:0) π σ,ϕ ( g − ) · h ( ϕ ) (cid:1) · m P,σ ( ϕ ) dϕ. A fundamental theorem of Harish-Chandra asserts that wave packets are Schwartz func-tions on G . Theorem 4.7.
The wave packets associated to the Schwartz-class functions from b A P into L ( Ind GP σ ) all belong to the Harish-Chandra Schwarz space S ( G ) . Moreover, The wave packet operator h → ˇ h is adjoint to the Fourier transform.Proof. See [21, Theorems 12.7.1 and 13.4.1] and [3, Corollary 9.8].
Let P = MAN be a parabolic subgroup. Suppose that π = index GP ( η M ⊗ ϕ ) where η M is anirreducible tempered representation of M with character denoted by Θ M ( η ) and ϕ ∈ b A P = b A ◦ × b A S . We denote r = dim ( b A S ) and n = dim ( b A ◦ ) . For i =
0, . . . , n , let h i ∈ S ( b A P ) , and v i , w i beunit K -finite vectors in Ind GP ( η M ) . Definition 4.8. If f i ∈ S ( G ) are wave packets associated to h i · v i ⊗ w ∗ i ∈ S (cid:0) b A P , L ( Ind GP ( η M ) (cid:1) , then we define a ( n + ) -linear map T π with image in S ( b A P ) by T π ( b f , . . . , b f n )= (cid:14)P τ ∈ S n sgn ( τ ) · h ( ϕ ) · Q ni = i ( ϕ ) ∂ τ ( i ) if v i = w i + , i =
0, . . . , n − and v n = w ; else . (4.1)Next we want to generalize the above definition to the Fourier transforms of all f ∈ S ( G ) .The induced space π = Ind GP ( η M ⊗ ϕ ) has a dense subspace: (cid:8) s : K → V η M continuous (cid:12)(cid:12) s ( km ) = η M ( m ) − s ( k ) for k ∈ K, m ∈ K ∩ M (cid:9) , (4.2)where V η M is the Hilbert space of M -representation η M . The group action on π is given bythe formula π ( g ) s ( k ) = e − h log ϕ + ρ,H ( g − k ) i · η M ( µ ( g − k )) − · s ( κ ( g − k )) , (4.3)where ρ = P α ∈ R + (a P , g) α . By Equation (4.3), the Fourier transform π ( f ) s ( k ) = b f ( π ) s ( k )= Z G ( e − h log ϕ + ρ,H ( g − k ) i · η M ( µ ( g − k )) − f ( g ) · s ( κ ( g − k )) dg. Suppose now that f , . . . , f n are arbitrary Schwartz functions on G and b f , . . . , b f n aretheir Fourier transforms. efinition 4.9. For any ≤ i ≤ n , we define a linear operator ∂∂ i on π = Ind GP ( η M ⊗ ϕ ) bythe following formula: (cid:16) ∂ b f ( π ) ∂ i (cid:17) s ( k ) := Z G H i ( g − k ) · ( e − h log ϕ + ρ,H ( g − k ) i · η M ( µ ( g − k )) − · f ( g ) · s ( κ ( g − k )) . (4.4)We define a ( n + ) -linear map T : S ( G ) → S ( b A P ) by T π ( b f , . . . b f n ) := X τ ∈ S n sgn ( τ ) · Trace (cid:16)b f ( π ) · n Y i = ∂ b f i ( π ) ∂ τ ( i ) (cid:17) . The above definition generalizes (4.1).
Proposition 4.10.
For any π = Ind GP ( η M ⊗ ϕ ) , we have the following identity: T π ( b f , . . . b f n )=(− ) n X τ ∈ S n sgn ( τ ) Z KMAN Z G × n H τ ( ) (cid:0) g . . . g n k (cid:1) . . . H τ ( n ) (cid:0) g n k (cid:1) e h log ϕ + ρ, log a i · Θ M ( η )( m ) · f ( kmank − ( g g . . . g n ) − ) f ( g ) . . . f n ( g n ) . (4.5) Proof.
By definition, for any τ ∈ S n , (cid:16)b f ( π ) · k Y i = ∂ b f i ( π ) ∂ τ ( i ) (cid:17) s ( k )= Z G × ( k + ) H τ ( ) ( g − κ ( g − k )) H τ ( ) (cid:0) g − κ (( g g ) − k ) (cid:1) H τ ( n ) (cid:0) g − κ (( g g . . . g n − ) − k ) (cid:1) · e − h log ϕ + ρ,H (( g g ...g n ) − k ) i η M ( µ (( g g . . . g n ) − k )) − · f ( g ) f ( g ) . . . f n ( g n ) s (cid:0) κ (( g g . . . g n ) − k ) (cid:1) . By setting g = ( g g . . . g n ) − k , we have g = kg − ( g g . . . g n ) − , and ( g g . . . g j ) − k = g j + g j + . . . g n g. We denote g − = mank ′ − ∈ MANK = G . Thus, b f ( π ) · k Y i = ∂ b f i ( π ) ∂ τ ( i ) ! s ( k )= Z KMAN Z G × k H τ ( ) (cid:0) g − κ ( g . . . g n k ) (cid:1) . . . H τ ( n ) (cid:0) g − κ ( g n k ) (cid:1) e h log ϕ + ρ, log a i · η M ( m ) · f (cid:16) kmank ′ − ( g g . . . g n ) − (cid:17) f ( g ) . . . f n ( g n ) s ( k ′ ) . (4.6)By Lemma 3.1, H τ ( i ) (cid:0) g − κ ( g i . . . g n k ) (cid:1) = H τ ( i ) (cid:0) g i + . . . g n k (cid:1) − H τ ( i ) (cid:0) g i . . . g n k (cid:1) . Thus, X τ ∈ S n sgn ( τ ) H τ ( ) (cid:0) g − κ ( g . . . g n k ) (cid:1) H τ ( ) (cid:0) g − κ (( g g . . . g n k ) (cid:1) . . . H τ ( n ) (cid:0) g − κ ( g n k ) (cid:1) = X τ ∈ S n sgn ( τ ) (cid:0) H τ ( ) ( g . . . g n k ) − H τ ( ) ( g g . . . g n k ) (cid:1)(cid:0) H τ ( ) ( g . . . g n k ) − H τ ( ) ( g . . . g n k ) (cid:1) . . . (cid:0) H τ ( n − ) ( g n k ) − H τ ( n − ) ( g n − g n k ) (cid:1)(cid:0) − H τ ( n ) ( g n k ) (cid:1) . (4.7) y induction on n , we can prove that the right hand side of equation (4.7) equals to (− ) n X τ ∈ S n sgn ( τ ) H τ ( ) (cid:0) g . . . g n k (cid:1) H τ ( ) (cid:0) g g . . . g n k (cid:1) · H τ ( n ) (cid:0) g n k (cid:1) . By (4.6) and (4.7), we conclude that X τ ∈ S n sgn ( τ ) (cid:16)b f ( π ) · k Y i = ∂ b f i ( π ) ∂ τ ( i ) (cid:17) s ( k )=(− ) n X τ ∈ S n sgn ( τ ) Z KMAN Z G × k H τ ( ) (cid:0) g . . . g n k (cid:1) H τ ( ) (cid:0) g g . . . g n k (cid:1) · H τ ( n ) (cid:0) g n k (cid:1) e h log ϕ + ρ, log a i · η M ( m ) · f (cid:0) kmank ′ − ( g g . . . g n ) − (cid:1) f ( g ) . . . f n ( g n ) s ( k ′ ) . Expressing it as a kernel operator, we have X τ ∈ S n sgn ( τ ) · (cid:16)b f ( π ) · k Y i = ∂ b f i ( π ) ∂ τ ( i ) (cid:17) s ( k ) = Z K L ( k, k ′ ) s ( k ′ ) dk ′ . where L ( k, k ′ ) =(− ) n X τ ∈ S n sgn ( τ ) Z MAN Z G × k H τ ( ) (cid:0) g . . . g n k (cid:1) H τ ( ) (cid:0) g g . . . g n k (cid:1) · H τ ( n ) (cid:0) g n k (cid:1) e h log ϕ + ρ, log a i · η M ( m ) · f ( kmank ′ − ( g g . . . g n ) − ) f ( g ) . . . f n ( g n ) . The proposition follows from the fact that T π = R K L ( k, k ) dk .Suppose that P and P are two non-conjugated parabolic subgroups such that Ind GP ( σ ⊗ ϕ ) can be embedded into Ind GP ( σ ⊗ ϕ ) , i.e.Ind GP (cid:0) σ ⊗ ϕ (cid:1) = M k Ind GP (cid:0) δ k ⊗ ϕ (cid:1) , where σ is a discrete series representation of M and δ k are different limit of discrete seriesrepresentations of M . We decompose b A = b A × b A = b A ◦ × b A S × b A . Let h i ∈ S ( b A ) , and v i , w i be unit K -finite vectors in Ind GP ( σ ) for i =
0, . . . n . We put b f i = h i · v i ⊗ w ∗ i ∈ S (cid:0) b A , L ( Ind GP ( σ ) (cid:1) . The following lemma follows from Definition 4.9.
Lemma 4.11.
Suppose that π = Ind GP (cid:0) δ k ⊗ ϕ (cid:1) . If v i = w i + , i =
0, . . . , n − and v n = w ∈ π , then T π ( b f , . . . , b f n ) = X τ ∈ S n sgn ( τ ) · h (cid:0) ( ϕ , 0 ) (cid:1) · n Y i = ∂h i (cid:0) ( ϕ , 0 ) (cid:1) ∂ τ ( i ) . Otherwise T π ( b f , . . . , b f n ) = . .4 Cocycles on b G temp Let P ◦ = M ◦ A ◦ N ◦ be a maximal parabolic subgroup and T be the maximal torus of K . Inparticular, T is the compact Cartan subgroup of M ◦ . Definition 4.12.
For an irreducible tempered representation π of G , we define A ( π ) = (cid:10) η M ◦ ⊗ ϕ ∈ ( \ M ◦ A ◦ ) temp (cid:12)(cid:12)(cid:12) Ind GP ◦ ( η M ◦ ⊗ ϕ ) = π (cid:11) . Definition 4.13.
Let m ( η M ◦ ) be the Plancherel density for the irreducible tempered repre-sentations η M ◦ of M ◦ . We put µ (cid:0) π (cid:1) = X η M ◦ ⊗ ϕ ∈ A ( π ) m ( η M ◦ ) . Recall the Plancherel formula f ( e ) = Z π ∈ b G temp Trace (cid:0)b f ( π ) (cid:1) · m ( π ) dπ, where m ( π ) is the Plancherel density for the G -representation π . Definition 4.14.
We define b Φ e by the following formula: b Φ e ( b f , . . . , s b f n ) = Z π ∈ b G temp T π ( b f , . . . , b f n ) · µ ( π ) · dπ. Theorem 4.15.
For any f , . . . f n ∈ S ( G ) , the following identity holds, Φ e ( f , . . . , f n ) = (− ) n b Φ e ( b f , . . . , b f n ) . The proof of Theorem 4.15 is presented in Section 4.5.
Example 4.16.
Suppose that G = R n . Let x i = ( x i1 , . . . x in ) ∈ R n be the coordinates of R n . On S ( R n ) , the cocycle Φ e is given as follows, Φ e ( f , . . . f n )= X τ ∈ S n sgn ( τ ) Z x ∈ R n · · · Z x n ∈ R n x ( ) . . . x nτ ( n ) f (cid:0) − ( x + · · · + x n ) (cid:1) f ( x ) . . . f n ( x n ) . On the other hand, the cocycle b Φ e on S ( c R n ) is given as follows, b Φ e ( b f , . . . , b f n ) = (− ) n Z R n b f d b f . . . d b f n . To see they are equal, we can compute Φ e ( f , . . . , f n ) = X τ ∈ S n sgn ( τ ) · (cid:16) f ∗ ( x τ ( ) f ) ∗ · · · ∗ ( x τ ( n ) f n ) (cid:17) ( ) . = X τ ∈ S n sgn ( τ ) · Z R n (cid:16) f ∗ (cid:0) x τ ( ) f (cid:1) ∗ · · · ∗ (cid:0) x τ ( n ) f n (cid:1) V (cid:17) = X τ ∈ S n sgn ( τ ) · Z R n (cid:16) b f · (cid:0) x τ ( ) f (cid:1) V . . . (cid:0) x τ ( n ) f n (cid:1) V (cid:17) = X τ ∈ S n sgn ( τ ) · Z R n (− ) n (cid:16) b f · ∂ b f ∂ τ ( ) . . . ∂ c f n ∂ τ ( n ) (cid:17) =(− ) n Z R n b f d b f . . . d b f n . o introduce the cocycle b Φ t for any t ∈ T reg , we first recall the formula of orbital integral(D.2) splits into three parts:regular part + singular part + higher part . Accordingly, for any t ∈ T reg , we define • regular part: for regular λ ∈ Λ ∗ K + ρ c (see Definition B.1), we define h b Φ t ( b f , . . . , b f n ) i λ = X w ∈ W K (− ) w · e w · λ ( t ) ! · Z ϕ ∈ b A ◦ T Ind GP ◦ ( σ M ◦ ( λ ) ⊗ ϕ ) ( b f , . . . , b f n ) · dϕ, where σ M ◦ ( λ ) is the discrete series representation of M ◦ with Harish-Chandra param-eter λ . • singular part: for any singular λ ∈ Λ ∗ K + ρ c , we define h b Φ t ( b f , . . . , b f n ) i λ = P w ∈ W K (− ) w · e w · λ ( t ) n ( λ ) · n ( λ ) X i = Z ϕ ∈ b A ◦ ǫ ( i ) · T Ind GP ◦ ( σ M ◦ i ( λ ) ⊗ ϕ ) ( b f , . . . , b f n ) · dϕ where σ M ◦ i are limit of discrete series representations of M ◦ with Harish-Chandraparameter λ organized as in Theorem D.3, and ǫ ( i ) = for i =
1, . . . n ( λ ) and ǫ ( i ) =− for i = n ( λ ) +
1, . . . n ( λ ) . • higher part: h b Φ t ( b f , . . . , b f n ) i high = Z π ∈ b G hightemp T π ( b f , . . . , b f n ) · X η M ◦ ⊗ ϕ ∈ A ( π ) κ M ◦ ( η M ◦ , t ) · dϕ, where the functions κ M ◦ ( η M ◦ , t ) are defined in Subsection D.2, and b G hightemp = (cid:8) π ∈ b G temp (cid:12)(cid:12) π = Ind GP ◦ ( η M ◦ ⊗ ϕ ) , η M ◦ ∈ ( c M ◦ ) hightemp (cid:9) . Definition 4.17.
For any element t ∈ T reg , we define b Φ t ( b f , . . . , b f n )= X regular λ ∈ Λ ∗ K + ρ c h b Φ h ( b f , . . . , b f n ) i λ + X singular λ ∈ Λ ∗ K + ρ c h b Φ t ( b f , . . . , b f n ) i λ + h b Φ t ( b f , . . . , b f n ) i high . Theorem 4.18.
For any t ∈ T reg , and f , . . . f n ∈ S ( G ) , the following identity holds, ∆ M ◦ T ( t ) · Φ t ( f , . . . , f n ) = (− ) n b Φ t ( b f , . . . , b f n ) . The proof of Theorem 4.18 is presented in Section 4.6. .5 Proof of Theorem 4.15 We split the proof into several steps:
Step 1 : Change the integral from b G temp to ( \ M ◦ A ◦ ) temp : b Φ e ( f , . . . , f n )= Z π ∈ b G temp T π ( b f , . . . , b f n ) · µ ( π ) · dπ = Z η M ◦ ⊗ ϕ ∈ ( \ M ◦ A ◦ ) temp T Ind GP ◦ ( η M ◦ ⊗ ϕ ) ( b f , . . . , b f n ) · m ( η M ◦ ) . Step 2 : Replace T Ind GP ◦ ( η M ◦ ⊗ ϕ ) in the above expression of b Φ e by Equation (4.5): b Φ e ( b f , . . . , b f n )=(− ) n X τ ∈ S n sgn ( τ ) Z η M ◦ ⊗ ϕ ∈ ( \ M ◦ A ◦ ) temp Z KM ◦ A ◦ N ◦ Z G × n H τ ( ) (cid:0) g . . . g n k (cid:1) . . . H τ ( n ) (cid:0) g n k (cid:1) e h log ϕ + ρ, log a i · Θ M ◦ ( η )( m ) · f (cid:0) kmank − ( g g . . . g n ) − (cid:1) f ( g ) . . . f n ( g n ) · m ( η M ◦ ) Step 3 : Simplify b Φ e by Harish-Chandra’s Plancherel formula. We write b Φ e ( b f , . . . , b f n )=(− ) n X τ ∈ S n sgn ( τ ) Z KN ◦ Z G × n H τ ( ) ( g . . . g n k ) . . . H τ ( n ) ( g n k ) · f ′ · f ( g ) . . . f n ( g n ) , where the function f ′ is defined the following formula f ′ = Z η M ◦ ⊗ ϕ ∈ ( \ M ◦ A ◦ ) temp Z M ◦ A ◦ e h log ϕ + ρ, log a i Θ M ◦ ( η )( m ) · f ( kmank − ( g g . . . g n ) − ) · m ( η M ◦ ) . If we put c = e h ρ, log a i · f (cid:0) kmank − ( g g . . . g n ) − (cid:1) , then f ′ = Z η M ◦ ⊗ ϕ ∈ ( \ M ◦ A ◦ ) temp (cid:0) Θ M ◦ ( η ) ⊗ ϕ (cid:1) ( c ) · m ( η M ◦ ) . By Theorem D.1, f ′ = c (cid:12)(cid:12) m = e,a = e = f (cid:0) knk − ( g g . . . g n ) − (cid:1) . This completes the proof.
Our proof strategy for Theorem 4.18 is similar to Theorem 4.15. We split its proof into 3steps as before.
Step 1 : Let Λ ∗ K ∩ M ◦ be the intersection of Λ ∗ T and the positive Weyl chamber for the group M ◦ ∩ K . We denote by ρ M ◦ ∩ Kc the half sum of positive roots in R (t , m ◦ ∩ k) . For any λ ∈ Λ ∗ K ∩ M ◦ , we can find an element w ∈ W K /W K ∩ M ◦ such that w · λ ∈ Λ ∗ K . Moreover, for any w ∈ W K /W K ∩ M ◦ , the induced representationInd GP ◦ ( σ M ◦ ( λ ) ⊗ ϕ ) ∼ = Ind GP ◦ ( σ M ◦ ( w · λ ) ⊗ ϕ ) . he regular part X regular λ ∈ Λ ∗ K + ρ c h b Φ t ( b f , . . . , b f n ) i λ = X regular λ ∈ Λ ∗ K + ρ c X w ∈ W K (− ) w · e w · λ ( t ) ! · Z ϕ ∈ b A ◦ T Ind GP ◦ ( σ M ◦ ( λ ) ⊗ ϕ ) ( b f , . . . , b f n ) · dϕ = X regular λ ∈ Λ ∗ K ∩ M ◦ + ρ c X w ∈ W K ∩ M ◦ (− ) w · e w · λ ( t ) · Z ϕ ∈ b A ◦ T Ind GP ◦ ( σ M ◦ ( λ ) ⊗ ϕ ) ( b f , . . . , b f n ) · dϕ. (4.8)Remembering that the above is anti-invariant under the W K -action, we can replace ρ c by ρ M ◦ ∩ Kc . That is, (4.8) equals to X regular λ ∈ Λ ∗ K ∩ M ◦ + ρ M ◦∩ Kc X w ∈ W K ∩ M ◦ (− ) w · e w · λ ( t ) · Z ϕ ∈ b A ◦ T Ind GP ◦ ( σ M ◦ ( λ ) ⊗ ϕ ) ( b f , . . . , b f n ) · dϕ. Similarly, the singular part X singular λ ∈ Λ ∗ K + ρ c h b Φ t ( b f , . . . , b f n ) i λ = X singular λ ∈ Λ ∗ K ∩ M ◦ + ρ M ◦∩ Kc X w ∈ W K ∩ M ◦ (− ) w · e w · λ ( t ) × n ( λ ) X i = ǫ ( i ) n ( λ ) · Z ϕ ∈ b A ◦ T Ind GP ◦ ( σ M ◦ i ( λ ) ⊗ ϕ ) ( b f , . . . , b f n ) . At last, the higher part h b Φ t ( b f , . . . , b f n ) i high = Z π ∈ b G hightemp T π ( b f , . . . , b f n ) (cid:0) X η M ◦ ⊗ ϕ ∈ A ( π ) κ M ◦ ( η M ◦ , t ) (cid:1) · dϕ = Z η M ◦ ⊗ ϕ ∈ c M hightemp × b A ◦ T Ind GP ◦ ( η M ◦ ⊗ ϕ ) ( b f , . . . , b f n ) · κ M ◦ ( η M ◦ , t ) . Step 2 : We apply Proposition 4.10 and obtain the following. • regular part: X regular λ ∈ Λ ∗ K + ρ c h b Φ t ( b f , . . . , b f n ) i λ =(− ) n X τ ∈ S n sgn ( τ ) X regular λ ∈ Λ ∗ K ∩ M ◦ + ρ M ◦∩ Kc X w ∈ W K ∩ M ◦ (− ) w · e w · λ ( t ) Z ϕ ∈ b A ◦ Z KM ◦ A ◦ N ◦ Z G × n H τ ( ) (cid:0) g . . . g n k (cid:1) . . . H τ ( n ) (cid:0) g n k (cid:1) e h log ϕ + ρ, log a i · Θ M ◦ (cid:0) λ (cid:1) ( m ) · f (cid:0) kmank − ( g g . . . g n ) − (cid:1) f ( g ) . . . f n ( g n ) . singular part: X singular λ ∈ Λ ∗ K + ρ c h b Φ t ( b f , . . . , b f n ) i λ =(− ) n X τ ∈ S n sgn ( τ ) X singular λ ∈ Λ ∗ K ∩ M ◦ + ρ M ◦∩ Kc n ( λ ) X i = ǫ ( i ) n ( λ ) X w ∈ W K ∩ M ◦ (− ) w · e w · λ ( t ) Z ϕ ∈ b A ◦ Z KM ◦ A ◦ N ◦ Z G × n H τ ( ) (cid:0) g . . . g n k (cid:1) . . . H τ ( n ) (cid:0) g n k (cid:1) e h log ϕ + ρ, log a i · Θ M ◦ i ( λ )( m ) · f (cid:0) kmank − ( g g . . . g n ) − (cid:1) f ( g ) . . . f n ( g n ) . • higher part: h b Φ t ( b f , . . . , b f n ) i high =(− ) n Z η M ◦ ⊗ ϕ ∈ c M hightemp × b A ◦ Z KM ◦ A ◦ N ◦ Z G × n H τ ( ) (cid:0) g . . . g n k (cid:1) . . . H τ ( n ) (cid:0) g n k (cid:1) e h log ϕ + ρ, log a i · Θ M ◦ (cid:0) η (cid:1) ( m ) · f (cid:0) kmank − ( g g . . . g n ) − (cid:1) f ( g ) . . . f n ( g n ) · κ M ◦ ( η M ◦ , t ) . Step 3 : Combining all the above computation together, we have b Φ t ( b f , . . . , b f n )= X regular λ ∈ Λ ∗ K + ρ c h b Φ t ( b f , . . . , b f n ) i λ + X singular λ ∈ Λ ∗ K + ρ c h b Φ t ( b f , . . . , b f n ) i λ + h b Φ t ( b f , . . . , b f n ) i high =(− ) n Z KN ◦ Z G × n f ′ · X τ ∈ S n sgn ( τ ) · H τ ( ) (cid:0) g . . . g n k (cid:1) . . . H τ ( n ) (cid:0) g n k (cid:1) · f ( g ) . . . f n ( g n ) ! , (4.9)where f ′ = X regular λ ∈ Λ ∗ K ∩ M ◦ + ρ M ◦∩ Kc X w ∈ W K ∩ M ◦ (− ) w · e w · λ ( t ) · Z ϕ ∈ b A ◦ (cid:0) Θ M ◦ ( λ ) ⊗ ϕ (cid:1) ( c )+ X singular λ ∈ Λ ∗ K ∩ M ◦ + ρ M ◦∩ Kc P w ∈ W K ∩ M ◦ (− ) w · e w · λ ( t ) n ( λ ) ! · n ( λ ) X i = ǫ ( i ) · Z ϕ ∈ b A ◦ (cid:16) Θ M ◦ i ( λ ) ⊗ ϕ (cid:17) ( c )+ Z η M ◦ ⊗ ϕ ∈ c M hightemp ⊗ b A ◦ (cid:0) Θ M ◦ ( η ) ⊗ ϕ (cid:1) ( c ) · κ M ◦ ( η M ◦ , t ) , where c = e h ρ, log a i · f (cid:0) kmank − ( g g . . . g n ) − (cid:1) . We then apply Theorem D.3 to the function c . Hence we obtain that f ′ = F Tc ( t ) = ∆ M ◦ T ( t ) · Z h ∈ M ◦ /T ◦ f (cid:0) khth − nk − ( g g . . . g n ) − (cid:1) . (4.10) ence, by (4.9) and (4.10), we conclude that b Φ t ( b f , . . . , b f n )= ∆ M ◦ T ( t ) · (− ) n X τ ∈ S n sgn ( τ ) Z h ∈ M ◦ /T ◦ Z KN ◦ Z G × n H τ ( ) (cid:0) g . . . g n k (cid:1) . . . H τ ( n ) (cid:0) g n k (cid:1) f (cid:0) khth − nk − ( g g . . . g n ) − (cid:1) f ( g ) . . . f n ( g n )=(− ) n ∆ M ◦ T ( t ) · Φ t ( f , . . . , f n ) . This completes the proof.
In this section, we study K ( C ∗ r ( G )) by computing its pairing with Φ t , t ∈ T reg and Φ e .And we construct a group isomorphism F : K ( C ∗ r ( G )) → Rep ( K ) . K ( C ∗ r ( G )) In Theorem C.4, we explain that the K -theory group of C ∗ r ( G ) is a free abelian group gener-ated by the following components, i.e. K ( C ∗ r ( G )) ∼ = M [ P,σ ] ess K (cid:0) K (cid:0) C ∗ r ( G ) [ P,σ ] (cid:1) ∼ = M λ ∈ Λ ∗ K + ρ c Z . (5.1)Let [ P, σ ] ∈ P ( G ) be an essential class corresponds to λ ∈ Λ ∗ K + ρ c . In this subsection,we construct a generator of K ( C ∗ r ( G )) associated to λ . We decompose b A P = b A S × b A ◦ anddenote r = dim b A S and b A ◦ = n . By replacing G with G × R , we may assume that n is even.Let V be an r -dimensional complex vector space and W an n -dimensional Euclideanspace. Take z = ( x , · · · , x r , y , . . . y n ) , x i ∈ C , y j ∈ R to be coordinates on V ⊕ W . Assume that the finite group ( Z ) r acts on W by simple reflec-tions. In terms of coordinates, ( x , · · · , x r , y , . . . , y n ) → ( ± x , · · · , ± x r , y , . . . , y n ) . Let us consider the the Clifford algebraClifford ( V ) ⊗ Clifford ( W ) together with the spinor module S = S V ⊗ S W . Here the spinor modules are equipped witha Z -grading: S + = S + V ⊗ S + W ⊕ S − V ⊗ S − W , S − = S + V ⊗ S − W ⊕ S − V ⊗ S + W . Let S ( V ) , S ( W ) , and S ( V ⊕ W ) be the algebra of Schwarz functions on V , W , and V ⊕ W .The Clifford action c ( z ) : S ± → S ∓ c ( z ) ∈ End ( S V ) ⊗ End ( S W ) . Let e , . . . e r − be a basis for S + V , e r − + , . . . e r be a basis for S − V , and f , . . . f n2 a basis for S W . We write c i,j,k,l ( z ) = h c ( z ) e i ⊗ f l , e j ⊗ f k i , 1 ≤ i, j ≤ r , 1 ≤ k, l ≤ n2 , nd define T := e − | z | · id S + e − | z | ( − e − | z | ) · c ( z ) | z | e − | z | c ( z ) ( − e − | z | ) · id S − − (cid:18) id S − (cid:19) , which is a r + n2 × r + n2 matrix: (cid:0) t i,j,k,l (cid:1) , 1 ≤ i, j ≤ r , 1 ≤ k, l ≤ n2 , with t i,j,k,l ∈ S ( V ⊕ W ) . Definition 5.1.
On the n -dimensional Euclidean space W , we can define B n = e − | y | · id S + W e − | y | ( − e − | y | ) · c ( y ) | y | e − | y | c ( y ) ( − e − | y | ) · id S − W − (cid:18) id S − W (cid:19) , which is a n2 × n2 matrix: (cid:0) b k,l (cid:1) , 1 ≤ k, l ≤ n2 , with b k,l ∈ S ( W ) . Then B n is the Bott generator in K ( C ( W )) ∼ = Z . Lemma 5.2.
If we restrict to W ⊂ V ⊕ W , that is x = , then we have that T (cid:12)(cid:12) x = = (cid:18) id S + V − id S − V (cid:19) ⊗ B n . Proof.
By definition, we have that • t i,j,k,l = e − z when e i , e j , f k , f l ∈ S + ; • t i,j,k,l = − e − z when e i , e j , f k , f l ∈ S − ; • t i,j,k,l = e − | z | ( − e − | z | ) · c i,j,k,l ( z ) | z | when e i , f k , ∈ S + and e j , f l , ∈ S − ; • t i,j,k,l = e − | z | · c i,j,k,l ( z ) when e i , f k , ∈ S − and e j , f l , ∈ S + .Moreover, the Clifford action c ( z ) = c ( x ) ⊗ + ⊗ c ( y ) ∈ End ( S V ) ⊗ End ( S W ) for z = ( x, y ) ∈ V ⊕ W . Thus, c i,j,k,l ( z ) (cid:12)(cid:12) x = = c k,l ( y ) . This completes the proof.On the other hand, the induced representation decomposesInd GP ( σ ⊗ ϕ ⊗ ) = r M i = Ind GP (cid:0) δ i ⊗ ϕ (cid:1) . By Equation (B.1), the characters of the limit of discrete series representations of δ i are allthe same up to a sign after restricting to a compact Cartan subgroup of M P . We can organizethe numbering so that δ i , i =
1, . . . 2 r − have the same character after restriction and δ i , i = r − , . . . 2 r share the same character. In particular, δ i with ≤ i ≤ r − and δ j with r − + ≤ j ≤ r have the opposite characters after restriction. e fix r unit K -finite vectors v i ∈ Ind GP (cid:0) δ i (cid:1) , and define S λ := (cid:0) t i,j,k,l · v i ⊗ v ∗ j (cid:1) . (5.2)The matrix S λ ∈ h S (cid:0) b A P , L ( index GP σ ) (cid:1)i W σ , and it is an idempotent. Recall that K (cid:0) Ind GP σ (cid:1) W σ ∼ (cid:0) C ( R ) ⋊ Z (cid:1) r ⊗ C ( R n ) . Definition 5.3.
We define Q λ ∈ M r + n ( S ( G )) to be the wave packet associated to S λ . Then [ Q λ ] is the generator in K (cid:0) C ∗ r ( G ) [ P,σ ] (cid:1) for [ P, σ ] . Let G be a connected, linear, real, reductive Lie group with maximal compact subgroup K .We denote by T the maximal torus of K , and P ◦ the maximal parabolic subgroup of G . Itfollows from Appendix C that for any λ ∈ Λ ∗ K + ρ c , there is generator [ Q λ ] ∈ K (cid:0) C ∗ r ( G ) (cid:1) . In Section 3, we defined a family of cyclic cocycles Φ e , Φ t ∈ HC (cid:0) S ( G ) (cid:1) for all t ∈ T reg . Theorem 5.4.
The index pairing between cyclic cohomology and K -theory HP even (cid:0) S ( G ) (cid:1) ⊗ K (cid:0) S ( G ) (cid:1) → C is given by the following formulas: • We have h Φ e , [ Q λ ] i = (− ) dim ( A ◦ ) | W M ◦ ∩ K | · X w ∈ W K m (cid:0) σ M ◦ ( w · λ ) (cid:1) , where σ M ◦ ( w · λ ) is the discrete series representation with Harish-Chandra parameter w · λ ,and m (cid:0) σ M ◦ ( w · λ ) (cid:1) is its Plancherel measure; • For any t ∈ T reg , we have that h Φ t , [ Q λ ] i = (− ) dim ( A ◦ ) P w ∈ W K (− ) w e w · λ ( t ) ∆ M ◦ T ( t ) . (5.3)The proof of Theorem 5.4 is presented in Sections 5.3 and 5.4. Corollary 5.5.
The index paring of [ Q λ ] and normalized higher orbital integral equals to thecharacter of the representation Ind GP ◦ ( σ M ◦ ( λ ) ⊗ ϕ ) at ϕ = . That is, * ∆ M ◦ T ∆ GT · Φ t , [ Q λ ] + = (− ) dim ( A ◦ ) Θ ( P ◦ , σ M ◦ ( λ ) , 1 )( t ) . roof. It follows from applying the character formula, Corollary B.6, to the right side ofEquation (5.3).
Remark 5.6.
If the group G is of equal rank, then the normalization factor is trivial. Andthe above corollary says that the orbital integral equals to the character of (limit of ) discreteseries representations. This result in the equal rank case is also obtained by Hochs-Wang in[11] using fixed point theorem and the Connes-Kasparov isomorphism. In contrast to theHochs-Wang approach, our proof is based on representation theory and does not use anygeometry of the homogenous space G/K or the Connes-Kasparov theory.We notice that though the cocycles Φ t introduced in Definition 3.3 are only defined forregular elements in T , Theorem 5.4 suggests that the pairing ∆ M ◦ T ( t ) h Φ t , [ Q λ ] i is a welldefined smooth function on T . This inspires us to introduce the following map. Definition 5.7.
Define a map F T : K ( C ∗ r ( G )) → C ∞ ( T ) by F T ([ Q λ ])( t ) : = (− ) dim ( A ◦ ) · ∆ M ◦ T ∆ KT · h Φ t , [ Q λ ] i . Let Rep ( K ) be the character ring of the maximal compact subgroup K . By the Weyl char-acter formula, for any irreducible K -representation V λ with highest weight λ , its characteris given by Θ λ ( t ) = P w ∈ W K (− ) w e w · ( λ + ρ c ) ( t ) ∆ KT ( t ) . Corollary 5.8.
The map F T : K ( C ∗ r ( G )) → Rep ( K ) is an isomorphism of abelian groups. In [4], we will use the above property of F T to show that F T is actually the inverse of theConnes-Kasparov Dirac index map, index : Rep ( K ) → K ( C ∗ r ( G )) . Remark 5.9.
The cyclic homology of the algebra S ( G ) is studied by Wassermann [24]. Wasser-mann’s result together with Corollary 5.8 show that higher orbit integrals Φ t , t ∈ H reg , gen-erate HP • ( S ( G )) . Actually, Definition 3.3 can be generalized to construct a larger class ofHochschild cocycles for S ( G ) for every parabolic subgroup (not necessarily maximal) withdifferent versions of Theorem I and II. We plan to investigate this general construction andits connection to the Plancherel theory in the near future. Suppose that λ ∈ Λ ∗ K + ρ c is regular and σ M ◦ ( λ ) is the discrete series representation of M ◦ with Harish-Chandra parameter λ . We consider the generator [ Q λ ] , the wave packetassociated to the matrix S λ introduced in (5.2), corresponding toInd GP ◦ ( σ M ◦ ( λ ) ⊗ ϕ ) , ϕ ∈ b A ◦ . According to Theorem 4.15, we have that (− ) n h Φ e , Q λ i = Z π ∈ b G temp T π Trace S λ ⊗ · · · ⊗ S λ | {z } n + · µ ( π )= Z b A ◦ T Ind GP ◦ ( σ M ◦ ( λ ) ⊗ ϕ ) · Trace S λ ⊗ · · · ⊗ S λ | {z } n + · µ (cid:16) Ind GP ( σ M ◦ ( λ ) ⊗ ϕ ) (cid:17) . y definition, µ (cid:16) Ind GP ◦ ( σ M ◦ ( λ ) ⊗ ϕ ) (cid:17) = µ (cid:16) Ind GP ◦ ( σ M ◦ ( λ ) ⊗ (cid:17) = X w ∈ W K /W K ∩ M ◦ m (cid:0) σ M ◦ ( w · λ ) (cid:1) = | W K ∩ M ◦ | · X w ∈ W K m (cid:0) σ M ◦ ( w · λ ) (cid:1) . Moreover, in the case of regular λ , S λ = [ B n · ( v ⊗ v ∗ )] , where B n is the Bott generator for K ( S ( b A ◦ )) and v is a unit K -finite vector in Ind GP ◦ ( σ M ◦ ( λ )) . By (4.1), Z b A ◦ T Ind GP ◦ ( σ ( λ ) ⊗ ϕ ) Trace (cid:0) S λ ⊗ · · · ⊗ S λ | {z } n + (cid:1) = h B n , b n i = where [ b n ] ∈ HC n ( S ( R n )) is the cyclic cocycle on S ( R n ) of degree n , c.f. Example 4.16. Weconclude that h Φ e , [ Q λ ] i = (− ) n | W K ∩ M ◦ | · X w ∈ W K m (cid:0) σ M ◦ ( w · λ ) (cid:1) . For the orbital integral, only the regular part will contribute. The computation is similaras above, and we conclude that h Φ t , Q λ i = (− ) n X w ∈ W K (− ) w e w · λ ( t ) ! · Z b A ◦ T Ind GP ◦ ( σ M ◦ ( λ ) ⊗ ϕ ) Trace S λ ⊗ · · · ⊗ S λ | {z } n + =(− ) n X w ∈ W K (− ) w e w · λ ( t ) . Suppose now that λ ∈ Λ ∗ K + ρ c is singular. We decompose b A P = b A ◦ × b A S , ϕ = ( ϕ , ϕ ) . We denote r = dim ( A S ) and n = dim ( A ◦ ) as before. In this case, we have thatInd GP ( σ M ⊗ ϕ ⊗ ) = r M i = Ind GP ◦ (cid:0) σ M ◦ i ⊗ ϕ (cid:1) , where σ M is a discrete series representation of M and σ M ◦ i , i =
1, ..., 2 r , are limit of discreteseries representations of M ◦ with Harish-Chandra parameter λ .Recall that the generator Q λ is the wave packet associated to S λ . The index paring equalsto (− ) n h Φ e , Q λ i = Z b A P T Ind GP ( σ M ⊗ ϕ ) Trace S λ ⊗ · · · ⊗ S λ | {z } n + · µ (cid:16) Ind GP ( σ M ⊗ ϕ ) (cid:17) . By the definition of µ , µ (cid:16) Ind GP ( σ M ⊗ ϕ ) (cid:17) = X η M ◦ ⊗ ϕ ∈ A ( ( Ind GP ( σ M ⊗ ϕ ) ) m ( η M ◦ ) . hus, the function µ (cid:0) Ind GP ( σ M ⊗ ϕ ) (cid:1) is constant with respect to ϕ ∈ b A ◦ . It follows from(4.1) that (− ) n h Φ e , Q λ i = Z b A S µ (cid:16) Ind GP ( σ M ⊗ ϕ ) (cid:17) · Z b A ◦ T Ind GP ( σ M ,ϕ ) Trace ( S λ ⊗ · · · ⊗ S λ | {z } n + ) = X τ ∈ S n X j = i ,...j n − = i n ,j n = i Z b A S (cid:18) µ (cid:0) Ind GP ( σ M ⊗ ϕ ) Z b A ◦ (− ) τ · s i ,j ( ϕ ) ∂s i ,j ( ϕ ) ∂ τ ( ) . . . ∂s i n ,j n ( ϕ ) ∂ τ ( n ) (cid:19) , where s i,j ∈ S ( b A ◦ × b A S ) with ≤ i, j ≤ r + n2 is the coefficient of the ( i, j ) -th entry inthe matrix S λ . We notice that the dimension of b A P is n + r . It follows from the Connes-Hochschild-Kostant-Rosenberg theorem that the periodic cyclic cohomology of the algebra S ( b A P ) is spanned by a cyclic cocycle of degree n + r . Accordingly, we conclude that h [ Φ e ] , Q λ i = because it equals to the pairing of the Bott element B n + r ∈ K ( b A P ) and a cyclic cocycle in HC ( b A P ) with degree only n < n + r .Next we turn to the index pairing of orbital integrals. In this singular case, it is clear thatthe regular part of higher orbital integrals will not contribute. For the higher part, (− ) n h [ Φ t ] high , Q λ i = Z ϕ ∈ b A P T Ind GP ( σ M ⊗ ϕ ) Trace S λ ⊗ · · · ⊗ S λ | {z } n + · X η M ◦ ⊗ ϕ ∈ A ( Ind GP ( σ M ⊗ ϕ )) κ M ◦ ( η M ◦ , t ) . Note that the function X η M ◦ ⊗ ϕ ∈ A ( Ind GP ( σ M ⊗ ϕ )) κ M ◦ ( η M ◦ , t ) is constant on ϕ ∈ b A ◦ . By (4.1), we can see that Z b A S X η M ◦ ⊗ ϕ ∈ A ( Ind GP ( σ M ⊗ ϕ ) ) κ M ◦ (cid:0) η M ◦ , t (cid:1) · Z b A ◦ T Ind GP ( σ M ⊗ ϕ ) Trace S λ ⊗ · · · ⊗ S λ | {z } n + = X τ ∈ S n X j = i ,...j n − = i n ,j n = i Z b A S X η M ◦ ⊗ ϕ ∈ A ( Ind GP ( σ M ⊗ ϕ )) κ M ◦ ( η M ◦ , t ) Z b A ◦ (− ) τ · s i ,j ( ϕ ) ∂s i ,j ( ϕ ) ∂ τ ( ) . . . ∂s i n ,j n ( ϕ ) ∂ τ ( n ) ! . We conclude that h [ Φ t ] high , Q λ i = because it equals to the paring of the Bott element B n + r ∈ K ( b A P ) and a cyclic cocycle on S ( b A P ) of degree n , which is trivial in HP even ( S ( b A P )) . or the singular part, by Schur’s orthogonality, we have h [ Φ t ] λ ′ , Q λ i = unless λ ′ = λ .When λ ′ = λ , Theorem 4.18 gives us the following computation, (− ) n h [ Φ t ] λ , Q λ i = r X w ∈ W K (− ) w e w · λ ( t ) ! · r X k = Z ϕ ∈ b A P ǫ ( k ) · T Ind GP ◦ ( σ M ◦ k ⊗ ϕ ) Trace S λ ⊗ · · · ⊗ S λ | {z } n + . For each fixed k , it follows from Lemma 5.2 and Lemma 4.11 that Z ϕ ∈ b A P T Ind GP ◦ ( σ M ◦ k ⊗ ϕ ) Trace S λ ⊗ · · · ⊗ S λ | {z } n + = Z ϕ ∈ b A P T Ind GP ◦ ( σ M ◦ k ⊗ ϕ ) Trace S λ ⊗ · · · ⊗ S λ | {z } n + (cid:12)(cid:12)(cid:12) ϕ = = X j = i ,...j n − = i n ,j n = i = k X τ ∈ S n Z ϕ ∈ b A ◦ (− ) τ · s i ,j ( ϕ ) ∂s i ,j ( ϕ ) ∂ τ ( ) . . . ∂s i ,j ( ϕ ) ∂ τ ( ) = (cid:14) h B n , b n i = =
1, . . . 2 r − − h B n , b n i = = r − +
1, . . . 2 r . Combining all the above together and the fact that ǫ ( k ) = for k =
1, . . . 2 r − and ǫ ( k ) = − for k = r − +
1, . . . 2 r , we conclude that h [ Φ t ] , Q λ i = h [ Φ t ] λ , Q λ i = (− ) n X w ∈ W K (− ) w e w · λ ( t ) . AppendixA Integration of Schwartz functions
Let a ⊆ p be the maximal abelian subalgebra of p and h = t ⊕ a be the most non-compactCartan subalgebra of g . Let u = k ⊕ i p and U be the compact Lie group with Lie algebra u .Take v ∈ a ∗ an integral weight. Let ˜ v ∈ t ∗ ⊕ i a ∗ be an integral weight so that its restriction˜ v (cid:12)(cid:12) i a ∗ = i · v . Let G C be the complexification of G . Suppose that V be a finite-dimensionalirreducible holomorphic representation of G C with highest weight ˜ v . Introduce a Hermitianinner product V so that U acts on V unitarily.We take u v to be a unit vector in the sum of the weight spaces for weights that restrictto v on a . Lemma A.1.
For any g ∈ G , we have that e h v,H ( g ) i = k g · u v k . Proof.
The proof is borrowed from [12, Proposition 7.17]. By the Iwasawa decomposition,we write g = kan with a = exp ( X ) and X ∈ a . Since u v is the highest vector for the actionof a , n annihilates u v . Thus, k gu v k = k kau v k = e h v,X i k ku v k = e h v,X i . The last equation follows from the fact that K ⊆ U acts on V in a unitary way. On the otherhand, we have that H ( g ) = X . This completes the proof. roposition A.2. There exists a constant C v > 0 such that h v, H ( g ) i ≤ C v · k g k , where k g k is the distance from g · K to e · K on G/K .Proof.
Since G = K exp (a + ) K , we write g = k ′ exp ( X ) k with X ∈ a + . By definition, k g k = k X k , and H ( g ) = H ( ak ) . By the above lemma, we have that e h v,H ( ak ) i = k ak · u v k . We decompose k · u v into the weight spaces of a -action. That is k · u v = n X i = c i · u i , where c i ∈ C , k c i k ≤ and u i is a unit vector in the weight spaces for weights that restrictsto λ i ∈ a ∗ . It follows that k ak · u v k = k n X i = c i · a · u i k≤ n X i = k a · u i k = n X i = e h λ i ,X i k u i k ≤ e C v ·k X k , (A.1)where C v = n · sup Y ∈ a , with k Y k = (cid:8) h λ i , Y i (cid:12)(cid:12) ≤ i ≤ n (cid:9) . This completes the proof.Now let us fix a parabolic subgroup P = MAN . To prove the integral in the definitionof Φ P,x defines a continuous cochain on S ( G ) , we consider a family of Banach subalgebras S t ( G ) , t ∈ [ ∞ ] , of C ∗ r ( G ) , which was introduced and studied by Lafforgue, [13, Definition4.1.1]. Definition A.3.
For t ∈ [ ∞ ] , let S t ( G ) be the completion of C c ( G ) with respect to thenorm ν t defined as follows, ν t ( f ) := sup g ∈ G (cid:10) ( + k g k )) t Ξ ( g ) − (cid:12)(cid:12) f ( g ) (cid:12)(cid:12) (cid:11) . Proposition A.4.
The family of Banach spaces {S t ( G ) } t ≥ satisfies the following properties.1. For every t ∈ [ ∞ ) , S t ( G ) is a dense subalgebra of C ∗ r ( G ) stable under holomorphic func-tional calculus.2. For ≤ t < t < ∞ , k f k t ≤ k f k t , for f ∈ S t ( G ) . Therefore, S ( G ) ⊂ S t ( G ) ⊂ S t ( G ) .3. There exists a number d > 0 such that the integral f → f P ( ma ) := Z KN F ( kmank − ) is a continuous linear map from S t + d ( G ) to S t ( MA ) for t ∈ [ ∞ ) . . There exists T > 0 such that the orbit integral f → Z G/Z G ( x ) f ( gxg − ) is a continuous linear functional on S t ( G ) for t ≥ T , ∀ x ∈ G .Proof. Property 1 is from [13, Proposition 4.1.2]; Property 2 follows from the definition ofthe norm ν t ; Property 3 follows from [7, Lemma 21]; Property 4 follows from [7, Theorem6] Theorem A.5.
For any f , . . . f n ∈ S T + d + ( G ) for t ≥ T , and x ∈ M , the following integral Z h ∈ M/Z M ( x ) Z KN Z G × n H ( g k ) . . . H n ( g n k ) f (cid:0) khxh − nk − ( g . . . g n ) − (cid:1) · f ( g ) . . . f n ( g n ) is finite, and defines a continuous n -linear functional on S d + T + ( G ) .Proof. We put ˜ f i ( g i ) = sup k ∈ K (cid:8) (cid:12)(cid:12) H i ( g i k ) f i ( g i ) (cid:12)(cid:12) (cid:9) . By Proposition A.2, we find constants C i > 0 so that | H i ( g i k ) | ≤ C i k g i k k = C i k g i k . It shows from Definition A.3 that ˜ f i belongs to S d + T ( G ) , i =
1, ...n . Thus, the integrationin (A.1) is bounded by the following Z h ∈ M/Z M ( x ) Z KN Z G × n (cid:12)(cid:12) f (cid:0) khxh − nk − ( g . . . g n ) − (cid:1) · ˜ f ( g ) . . . ˜ f n ( g n ) (cid:12)(cid:12) = Z h ∈ M/Z M ( x ) Z KN F ( khxh − nk − ) (A.2)where by Prop. A.4.2 F = (cid:12)(cid:12) f ∗ ˜ f ∗ · · · ∗ ˜ f n (cid:12)(cid:12) ∈ S d + T ( G ) . For any m ∈ M, a ∈ A , we introduce F ( P ) ( ma ) = Z KN F ( kmank − ) . By Prop. A.4.3, we have that F ( P ) belongs to S T ( MA ) . Applying Prop. A.4.4 to the group MA , we conclude the orbital integral Z M/Z M ( x ) F ( P ) ( hxh − ) < + ∞ , from which we obtain the desired finiteness of the integral (A.2). Furthermore, with thecontinuity of the above maps, f i → ˜ f i , f ⊗ ˜ f ⊗ ... ⊗ ˜ f n → F, F → F ( P ) , F ( P ) → Z M/Z M ( x ) F ( P ) ( hxh − ) , we conclude that the integral (A.2) is a continuous n -linear functional on S d + T + ( G ) . Characters of representations of G B.1 Discrete series representation of G Suppose that rank G = rank K . Then G has a compact Cartan subgroup T with Lie algebradenoted by t . We choose a set of positive roots: R + (t , g) = R + c (t , g) ∪ R + n (t , g) , and define ρ c = X α ∈ R + c (t , g) α, ρ n = X α ∈ R + n (t , g) α, ρ = ρ c + ρ n . The choice of R + c (t , k) determines a positive Weyl chamber t ∗ + . Let Λ ∗ T be the weight latticein t ∗ . Then the set Λ ∗ K = Λ ∗ T ∩ t ∗ + parametrizes the set of irreducible K -representations. In addition, we denote by W K theWeyl group of the compact subgroup K . For any w ∈ W K , let l ( w ) be the length of w andwe denote by (− ) w = (− ) l ( w ) . Definition B.1.
Let λ ∈ Λ ∗ K + ρ c . We say that λ is regular if h λ, α i 6 = for all α ∈ R n (t , g) . Otherwise, we say λ is singular .Assume that q = dim G/K2 and T reg ⊂ T the set of regular elements in T . Theorem B.2 (Harish-Chandra) . For any regular λ ∈ Λ ∗ K + ρ c , there is a discrete series rep-resentation σ ( λ ) of G with Harish-Chandra parameter λ . Its character is given by the followingformula: Θ ( λ ) (cid:12)(cid:12) T reg = (− ) q · P w ∈ W K (− ) w e wλ ∆ GT , where ∆ GT = Y α ∈ R + (t , g) ( e α2 − e − α2 ) . Next we consider the case when λ ∈ Λ ∗ K + ρ c is singular. That is, there exists at least onenoncompact root α so that h λ, α i = . Choose a positive root system R + (t , g) that makes λ dominant; the choices of R + (t , g) are not unique when λ is singular. For every choice of R + (t , g) , we can associate it with a representation, denoted by σ (cid:0) λ, R + (cid:1) . We call σ ( λ, R + ) a limit of discrete series representation of G . Distinct choices of R + (t , g) lead to infinitesimallyequivalent versions of σ (cid:0) λ, R + (cid:1) . Let Θ (cid:0) λ, R + (cid:1) be the character of σ (cid:0) λ, R + (cid:1) . Then Θ (cid:0) λ, R + (cid:1)(cid:12)(cid:12) T reg = (− ) ± P w ∈ W K (− ) w e wλ ∆ GT . Moreover, for any w ∈ W K which fixes λ , we have that Θ (cid:0) λ, w · R + (cid:1)(cid:12)(cid:12) T reg = (− ) w · Θ (cid:0) λ, w · R + (cid:1)(cid:12)(cid:12) T reg . (B.1)See [12, P. 460] for more detailed discussion. .2 Discrete series representations of M Let P = MAN be a parabolic subgroup. The subgroup M might not be connected in general.We denote by M the connected component of M and set M ♯ = M Z M , where Z M is the center for M .Let σ be a discrete series representation (or limit of discrete series representation) forthe connected group M and χ be a unitary character of Z M . If σ has Harish-Chandraparameter λ , then we assume that χ (cid:12)(cid:12) T M ∩ Z M = e λ − ρ M (cid:12)(cid:12) T M ∩ Z M . We have the well-defined representation σ ⊠ χ of M ♯ , given by σ ⊠ χ ( gz ) = σ ( g ) χ ( z ) , for g ∈ M and z ∈ Z M . Definition B.3.
The discrete series representation or limit of discrete series representation σ for the possibly disconnected group M induced from σ ⊠ χ is defined as σ = Ind MM ♯ (cid:0) σ ⊠ χ (cid:1) . Discrete series representations of M are parametrized by a pair of Harish-Chandra pa-rameter λ and unitary character χ . Next we show that χ is redundant for the case of M ◦ .Denote • a = the Lie algebra of A ; • t M = the Lie algebra of the compact Cartan subgroup of M ; • a M = the maximal abelian subalgebra of p ∩ m , where g = k ⊕ p ;Then t M ⊕ a is a Cartan subalgebra of g , and a p = a M ⊕ a is a maximal abelian subalgebrain p .Let α be a real root in R (g , t M ⊕ a) . Restrict α to a and extend it by on a M to obtain arestricted root in R (g , a p ) . Form an element H α ∈ a p by the following α ( H ) = h H, H α i , H ∈ a p . It is direct to check that γ α = exp (cid:16) α | α | (cid:17) is a member of the center of M . Denote by F M the finite group generated by all γ α inducedfrom real roots of ∆ (g , t M ⊕ a) . It follows from Lemma 12.30 in [12] that M ♯ = M F M . (B.2) Lemma B.4.
For the maximal parabolic subgroup P ◦ = M ◦ A ◦ N ◦ , we have that Z M ◦ ⊆ ( M ◦ ) . Proof.
There is no real root in R (h ◦ , g) since the Cartan subgroup H ◦ is maximally compact.The lemma follows from (B.2).It follows that discrete series or limit of discrete series representation of M ◦ are para-metrized by Harish-Chandra parameter λ . We denote them by σ ( λ ) or σ ( λ, R + ) . .3 Induced representations of G Let P = MAN be a parabolic subgroup of G and L = MA its Levi subgroup as before.For any Cartan subgroup J of L , let { J , J , . . . , J k } be a complete set of representatives fordistinct conjugacy classes of Cartan subgroups of L for which J i is conjugate to J in G . Let x i ∈ G satisfy J i = x i Jx − and for j ∈ J , write j i = x i jx − . Theorem B.5.
Let Θ ( P, σ, ϕ ) be the character of the basic representation Ind GP ( σ ⊗ ϕ ) . Then • Θ ( P, σ, ϕ ) is a locally integrable function. • Θ ( P, σ, ϕ ) is nonvanishing only on Cartan subgroups of G that are G -conjugate to Cartansubgroups of L . • For any j ∈ J , we have Θ ( P, σ, ϕ )( j ) = k X i = | W ( J i , L ) | − | ∆ GJ i ( j i ) | − × (cid:16) X w ∈ W ( J i ,G ) | ∆ LJ i ( wj i ) | · Θ Mσ (cid:0) wj i (cid:12)(cid:12) M (cid:1) ϕ ( wj i | H p ) (cid:17) , (B.3) where Θ Mσ is the character for the M P representation σ , and the definition of ∆ GJ i (and ∆ LJ i ) isexplained in Theorem B.2.Proof. The first two properties of Θ ( P, σ, ϕ ) can be found in [12][Proposition 10.19], and thelast formula has been given in [10][Equation (2.9)]. Corollary B.6.
Suppose that P ◦ is the maximal parabolic subgroup of G and σ M ◦ ( λ ) is a (limit of)discrete series representation with Harish-Chandra parameter λ . We have that Θ (cid:0) P ◦ , σ M ◦ ( λ ) , ϕ (cid:1) ( h ) = P w ∈ K (− ) w e wλ ( h k ) · ϕ ( h p ) ∆ GH ◦ ( h ) . for any h ∈ H reg ◦ .Proof. The corollary follows from (B.3) and Theorem B.2.
C Description of K ( C ∗ r ( G )) Without loss of generality, we assume that dim A ◦ = n is even. Otherwise, we can replace G by G × R . C.1 Generalized Schmid identity
Suppose that P = MAN is a parabolic subgroup of G and H = TA is its associated Cartansubgroup. We assume that P is not maximal and thus H is not the most compact. By Cayleytransform, we can obtain a more compact Cartan subgroup H ∗ = T ∗ A ∗ . We denote by P ∗ = M ∗ A ∗ N ∗ the corresponding parabolic subgroup. Here A = A ∗ × R .Let σ be a (limit of) discrete series representation of M , and ν ⊗ ∈ b A = c A ∗ × b R . Suppose that π = Ind GP (cid:0) σ ⊗ ( ν ⊗ ) (cid:1) s a basic representation. Then π is either irreducible or decomposes as follows,Ind GP (cid:0) σ ⊗ ( ν ⊗ ) (cid:1) = Ind GP ∗ ( δ ⊗ ν ) ⊕ Ind GP ∗ ( δ ⊗ ν ) . Here δ and δ are limit of discrete series representations of M ∗ . Moreover, they share thesame Harish-Chandra parameter but corresponds to different choices of positive roots. Onthe right hand side of the above equation, if P ∗ is not maximal, then one can continue thedecomposition for Ind GP ∗ ( σ ∗ i ⊗ ν ) , i =
1, 2 . Eventually, we getInd GP (cid:0) σ ⊗ ( ϕ ⊗ ) (cid:1) = M i Ind GP ◦ ( δ i ⊗ ϕ ) , (C.1)where ϕ ⊗ ∈ b A P = b A ◦ × b A S . The number of component in the above decomposition is closely related to the R -groupwhich we will discuss below. We refer to [12, Corollary 14.72] for detailed discussion.As a consequence, we obtain the following lemma immediately. Lemma C.1.
Let P ◦ = M ◦ A ◦ N ◦ be the maximal parabolic subgroup. If σ ⊗ ϕ is an irreduciblerepresentation of M ◦ A ◦ , then the induced representation Ind GP ◦ (cid:0) σ ⊗ ϕ (cid:1) is also irreducible. C.2 Essential representations
Clare-Crisp-Higson proved in [3][Section 6] that the group C ∗ -algebra C ∗ r ( G ) has the fol-lowing decomposition: C ∗ r ( G ) ∼ = G [ P,σ ] ∈ P ( G ) C ∗ r ( G ) [ P,σ ] , where C ∗ r ( G ) [ P,σ ] ∼ = K (cid:0) Ind GP ( σ ) (cid:1) W σ . For principal series representations Ind GP ( σ ⊗ ϕ ) , Knapp and Stein [12, Chapter 9] showedthat the stabilizer W σ admits a semidirect product decomposition W σ = W ′ σ ⋊ R σ , where the R-group R σ consists of those elements that actually contribute nontrivially to theintertwining algebra of Ind GP ( σ ⊗ ϕ ) . Wassermann notes the following Morita equivalence, K (cid:0) Ind GP ( σ ) (cid:1) W σ ∼ C ( b A P /W ′ σ ) ⋊ R σ . Definition C.2.
We say that an equivalence class [ P, σ ] is essential if W σ = R σ . We denote itby [ P, σ ] ess . In this case, W σ = R σ ∼ = ( Z ) r is obtained by application of all combinations of r = dim ( A P ) − dim ( A ◦ ) commuting reflec-tions in simple noncompact roots.As before, let T be the maximal torus of K . We denote by Λ ∗ T and Λ ∗ K the weight latticeand its intersection with the positive Weyl chamber of K . Theorem C.3. (Clare-Higson-Song-Tang-Vogan) There is a bijection between the set of [ P, σ ] ess andthe set Λ ∗ K + ρ c . Moreover, Suppose that [ P, σ ] is essential. If λ is regular, that is, h λ, α i 6 = for all non-compact roots α ∈ R n , then W σ is trivial, P = P ◦ , and σ is the discrete seriesrepresentation of M ◦ with Harish-Chandra parameter λ . In addition, Ind GP ◦ ( σ ⊗ ϕ ) are irreducible for all ϕ ∈ b A ◦ . • Otherwise, if h λ, α i = for some α ∈ R n , then Ind GP ( σ ⊗ ϕ ⊗ ) = r M i = Ind GP ◦ (cid:0) δ i ⊗ ϕ (cid:1) , (C.2) where δ i is a limit of discrete series representation of M ◦ with Harish-Chandra parameter λ , ϕ ∈ b A ◦ and ϕ ⊗ ∈ b A P . The computation of K -theory group of C ∗ r ( G ) can be summarized as follows. Theorem C.4. (Clare-Higson-Song-Tang) The K -theory group of C ∗ r ( G ) is a free abelian groupgenerated by the following components, i.e. K ( C ∗ r ( G )) ∼ = M [ P,σ ] ess K (cid:0) K (cid:0) C ∗ r ( G ) [ P,σ ] (cid:1) ∼ = M [ P,σ ] ess K (cid:16) K (cid:0) Ind GP σ (cid:1) W σ (cid:17) ∼ = M regular part K (cid:0) C ( R n ) (cid:1) ⊕ M singular part K (cid:16)(cid:0) C ( R ) ⋊ Z (cid:1) r ⊗ C ( R n ) (cid:17) ∼ = M λ ∈ Λ ∗ K + ρ c Z . (C.3) Example C.5.
Let G = SL ( R ) . The principal series representations of SL ( R ) are para-metrized by characters ( σ, λ ) ∈ d MA ∼ = { ± } × R modulo the action of the Weyl group Z . One family of principal series representations isirreducible at while the other decomposes as a sum of two limit of discrete series repre-sentations. At the level of C ∗ r ( G ) , this can be explained as d MA/ Z ∼ = { + } × [ ∞ ) ∪ { − } × [ ∞ ) ∼ = { + } × R / Z ∪ { − } × R / Z , and the principal series contribute summands to C ∗ r ( SL ( R )) of the form C ( R / Z ) and C ( R ) ⋊ Z up to Morita equivalence. In addition SL ( R ) has discrete series representations each ofwhich contributes a summand of C to C ∗ r ( SL ( R )) , up to Morita equivalence. We obtain: C ∗ r ( SL ( R )) ∼ C ( R / Z ) ⊕ C ( R ) ⋊ Z ⊕ M n ∈ Z \{ } C . Here the part C ( R / Z ) corresponds to the family of spherical principal series representa-tions, which are not essential. Then (C.3) can be read as follows, K ( C ∗ r ( SL ( R ))) ∼ = K (cid:16)(cid:0) C ( R ) ⋊ Z (cid:1)(cid:17) ⊕ M n = K (cid:0) C (cid:1) . Orbital integrals
In this section, we assume that rank G = rank K . We denote by T the compact Cartansubgroup of G . D.1 Definition of orbital integrals
Let h be any semisimple element of G , and let Z G ( h ) denote its centralizer in G . Associatedwith h is an invariant distribution Λ f ( h ) given for f ∈ S ( G ) by f → Λ f ( h ) = Z G/Z G ( h ) f ( ghg − ) d G/Z G ( h ) ˙ g, where d ˙ g denotes a left G -invariant measure on the quotient G/Z G ( h ) . In this paper, we areonly interested in two cases: h = e and h is regular.If h = e is the identity element, the formula for Λ f ( e ) is the Harish-Chandra’s Plancherelformula for G . Theorem D.1 (Harish-Chandra’s Plancherel formula) . For any f ∈ S ( G ) , Λ f ( e ) = f ( e ) = Z π ∈ b G temp Θ ( π )( f ) · m ( π ) dπ where m ( π ) is the Plancherel density and Θ ( π ) is the character for the irreducible tempered repre-sentation π . Suppose that h is regular, then Z G ( h ) = H is a Cartan subgroup of G . We define ∆ GH = Y α ∈ R + (h , g) ( e α2 − e − α2 ) . We can normalize the measures on
G/H and H so that Z G f ( g ) dg = Z G/H Z H f ( gh ) dhd G/H ˙ g. Definition D.2.
For any t ∈ T reg , the orbital integral is defined by F Tf ( t ) = ∆ GT ( t ) Z G/T f ( gtg − ) d G/T ˙ g. Similarly, if h ∈ H reg , we define F Hf ( h ) = ǫ H ( h ) · ∆ GH ( h ) Z G/H f ( ghg − ) d G/H ˙ g, where ǫ H ( h ) is a sign function defined in [12, P. 349].Note that F Hf is anti-invariant under the Weyl group action, that is, for any element w ∈ W ( H, G ) , F Hf ( w · h ) = (− ) w · F Hf ( h ) . .2 The formula for orbital integrals In this subsection, we summarize the formulas and results in [9, 10]. If P is the minimalparabolic subgroup with the most non-compact Cartan subgroup H , then the Fourier trans-form of orbital integral equals to the character of representation. That is, for any h ∈ H reg b F Hf ( h ) = Z χ ∈ b H χ ( h ) · F Hf ( h ) · dχ = Θ ( P, χ )( f ) or equivalently, F Hf ( h ) = Z χ ∈ b H Θ ( P, χ )( f ) · χ ( h ) · dχ. For any arbitrary parabolic subgroup P , the formula for orbital integral is much more com-plicated, given as follows, F Hf ( h ) = X Q ∈ Par ( G,P ) Z χ ∈ b J Θ ( Q, χ )( f ) · κ G ( Q, χ, h ) dχ, (D.1)where • the sum ranges over the setPar ( G, P ) = (cid:8) parabolic subgroup Q of G (cid:12)(cid:12) Q is no more compact than P (cid:9) . • J is the Cartan subgroup associated to the parabolic subgroup Q ; • χ is a unitary character of J and Θ ( Q, χ ) is a tempered invariant eigen-distribution de-fined in [9]. In particular, Θ ( Q, χ ) is the character of parabolic induced representationor an alternating sum of characters which can be embedded in a reducible unitaryprincipal series representation associated to a different parabolic subgroup; • The function κ G is rather complicated to compute. Nevertheless, for the purpose ofthis paper, we only need to know the existence of functions κ G , which has been veri-fied in [19].In a special case when P = G and H = T , the formula (D.1) has the following moreexplicit form. Theorem D.3.
For any t ∈ T reg , the orbital integral F Tf ( t ) = X regular λ ∈ Λ ∗ K + ρ c X w ∈ W K (− ) w · e w · λ ( t ) · Θ ( λ )( f )+ X singular λ ∈ Λ ∗ K + ρ c X w ∈ W K (− ) w · e w · λ ( t ) · Θ ( λ )( f )+ Z π ∈ b G hightemp Θ ( π )( f ) · κ G ( π, t ) dχ. (D.2) In the above formula, there are three parts: • regular part: Θ ( λ ) is the character of the discrete series representation with Harish-Chandraparameter λ ; • singular part: for singular λ ∈ Λ ∗ K + ρ c , we denote by n ( λ ) the number of different limit ofdiscrete series representations with Harish-Chandra parameter λ . By (B.1), we can organizethem so that Θ ( λ ) (cid:12)(cid:12) T reg = · · · = Θ n ( λ ) ( λ ) (cid:12)(cid:12) T reg = − Θ n ( λ ) + ( λ ) (cid:12)(cid:12) T reg = · · · = − Θ n ( λ ) ( λ ) (cid:12)(cid:12) T reg . e put Θ ( λ ) : = ( λ ) · (cid:16) n ( λ ) X i = Θ i ( λ ) − n ( λ ) X i = n ( λ ) + Θ i ( λ ) (cid:17) . • higher part: b G hightemp is a subset of b G temp consisting of irreducible tempered representations whichare not (limit of) discrete series representations. References [1] Dan Barbasch and Henri Moscovici. L -index and the Selberg trace formula. J. Funct.Anal. , 53(2):151–201, 1983.[2] Philippe Blanc and Jean-Luc Brylinski. Cyclic homology and the Selberg principle.
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