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Higher Orbit Integrals, Cyclic Cocyles, and K-Theory of Reduced Group C

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Higher Orbit Integrals, Cyclic Cocyles, and K-Theory of Reduced Group C Higher Orbit Integrals, Cyclic Cocyles, and K-theory of Reduced Group C∗-algebra Yanli Song,∗ Xiang Tang† Abstract Let G be a connected real reductive group. Orbit integrals define traces on the group algebra of G. We introduce a construction of higher orbit integrals in the direction of higher cyclic cocycles on the Harish-Chandra Schwartz algebra of G. We analyze these higher orbit integrals 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 representation theory and K-theory. Contents 1 Introduction 2 2 Preliminary 5 2.1 Reductive Lie group and Cartan subgroups . ....... 5 2.2 Harish-Chandra’sSchwartzfunction . ........ 6 2.3 Cycliccohomology ................................ .. 7 3 Higher orbit integrals 7 3.1 Highercycliccocycles .. .... .... ... .... .... .... ... .... 7 3.2 Cocyclecondition................................ ... 9 3.3 Cycliccondition................................. ... 12 4 The Fourier transform of Φx 14 arXiv:1910.00175v2 [math.KT] 7 Nov 2019 4.1 Parabolicinduction .... .... .... ... .... .... .... ... .... 14 4.2 Wavepacket...................................... 15 4.3 DerivativesofFouriertransform . ....... 16 4.4 Cocycles on Gtemp ................................... 19 4.5 ProofofTheorem4.15 .............................. .. 21 4.6 ProofofTheorem4.18b .............................. .. 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]. 1 5 Higher Index Pairing 24 ∗ 5.1 Generators of K(Cr(G)) ................................ 24 5.2 Themainresults.................................. .. 26 5.3 Regularcase ..................................... 27 5.4 Singularcase.................................... .. 28 A Integration of Schwartz functions 30 B Characters of representations of G 33 B.1 Discrete series representation of G ......................... 33 B.2 Discrete series representations of M ........................ 34 B.3 Induced representations of G ............................ 35 ∗ C Description of K(Cr(G)) 35 C.1 GeneralizedSchmididentity. ...... 35 C.2 Essentialrepresentations . ....... 36 D Orbital integrals 38 D.1 Definition of orbital integrals . ....... 38 D.2 Theformulafororbitalintegrals . ....... 39 1 Introduction Let G be a connected real reductive group, and H a Cartan subgroup of G. Let f be a compactly supported smooth function on G. For x ∈ Hreg, the integrals H −1 Λf (x) := f(gxg )dG/Hg˙ ZG/ZG(x) 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 and its generalizations. Let W(H, G) be the subgroup of the Weyl group of G consisting of elements fixing H. Following Harish-Chandra, we define the orbit integral associated to H to be H ∞ reg −W(H,G) H H G −1 F : S(G) C (H ) , Ff (x) := ǫ (x)∆H(x) f(gxg )dG/Hg,˙ ZG/ZG(x) where C∞(Hreg)→−W(H,G) is the space of anti-symmetric functions with respect to the Weyl H G group W(H, G) action on H, ǫ (h) is a sign function on H, and ∆H is the Weyl denominator for H. Our starting point is the following property that for a given h ∈ Hreg, the linear functional on S(G), H H F (h): f 7 Ff (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 between periodic cyclic cohomology and K-theory of an algebra. The pairing between the orbit inte- H grals F (h) and K0(S(G)) behaves differently between the cases when G is of equal rank and non-equal rank. More explicitly, we will show in this article that when G has equal rank, FH 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-equal rank, FH vanishes on K-theory of S(G) completely, c.f. [11]. Furthermore, many numerical 2 invariants for G-equivariant Dirac operators in literature, e.g . [1, 5, 11, 14, 22] etc, vanish when G has non-equal rank. Our main goal in this article is to introduce generalizations of orbit integrals in the sense of higher cyclic cocycles on S(G) which will treat equal and non-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 from prior works in its emphasis on explicit cocycles. To understand the non-equal rank case better, we start with the example of the abelian group G=R, which turns out to be very instructive. Here S(R) is the usual algebra of Schwartz functions on R with convolution product, and it carries a nontrivial degree one cyclic cohomology. Indeed we can define a cyclic cocycle ϕ on S(R) as follows, (c.f. [16, Prop. 1.4]: ϕ(f0,f1)= xf0(−x)f1(x)dx. (1.1) R Z Under the Fourier transform, the convolution algebra S(R) is transformed into the Schwartz functions with pointwise multiplication, and the cocycle ϕ is transformed into something more familiar: ϕ^(f^0, f^1)= f^0df^1. (1.2) Rb Z It follows from the Hochschild-Kostant-Rosenberg theorem that ϕ^ generates the degree one cyclic cohomology of S(R), and accordingly ϕ generates the degree one cyclic cohomology of S(R). We notice that it is crucialb to have the function x in Equation (1.1) to have the integral of f^0df^1 on S(R). And our key discovery is a natural generalization of the function x on a general connected real reductive group G. Let P = MAN be a parabolic subgroup of G. The Iwasawa decompositionb G = KMAN writes an element g ∈ G as g = κ(g)µ(g)eH(g)n ∈ KMAN = G. Let dim(A)= n. And the function H = (H1,...,Hn): G a provides us the right ingredi- ent to generalize the cocycle ϕ in Equation (1.1). We introduce a generalization ΦP for orbit integrals in Definition 3.3. For f0, ..., fn ∈ S(G) and x ∈ M→, ΦP,x is defined by the following integral, ΦP,x(f0,f1,...,fn) := sgn(τ) · Hτ(1)(g1...gnk)Hτ(2)(g2...gnk) ...Hτ(n)(gnk) ×n h∈M/ZM (x) KN G Z Z Z τX∈Sn −1 −1 −1 f0 khxh nk (g1 ...gn) f1(g1) ...fn(gn)dg1 ··· dgndkdndh, where ZM(x) is the centralizer of xin 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, which leads 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◦, the cochain ΦP◦,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 orbit integral ΦP 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 Φx defined as an integral on G. The following theorem in Section 4 establishes a full generalization of Equation (1.2) to connected real reductive groups. b b 3 reg Theorem II. (Theorem 4.15 and 4.18) Let T be a compact Cartan subgroup of M◦. For any t ∈ T , and f0,...fn ∈ S(G), the following identity holds, n ΦP◦,e(f0,...,fn)=(−1) Φe(f0,..., fn), M◦ n ∆ (t) · ΦP◦,t(f0,...,fn)=(−1) Φt(f0,..., fn), T b b b M◦ where f0, ..., fn are the Fourier transforms of f0, ··· ,fn ∈ Sb(G)b, and ∆Tb (t) is the Weyl denomi- nator of T. b b As an application of our study, we compute the pairing between the K-theory of S(G) and ΦP. Vincent Lafforgue showed in [13] that Harish-Chandra’s Schwartz algebra S(G) ∗ is a subalgebra of Cr(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 Cr(G). The ∗ structure of Cr(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 Cr(G) as a free abelian group, c.f. Theorem C.3. With ∗ wave packet, we construct a representative [Qλ] ∈ K(Cr(G)) for each generator in Theorem C.3. Applying Harish-Chandra’s theory of orbit integrals, we compute explicitly in the fol- lowing theorem the index pairing between [Qλ] and ΦP. Theorem III. (Theorem 5.4) The index pairing between cyclic cohomology and K-theory even HP S(G) ⊗ K0 S(G) C is given by the following formulas: → • We have dim(AP) (−1) M◦ hΦe, [Qλ]i = · m σ (w · λ) , |WM◦∩K| w∈W XK where σM◦ (w· λ) is the discrete series representation with Harish-Chandra parameter w · λ, and m σM◦ (w · λ) is its Plancherel measure; reg • For anyt ∈ T , we have that (−1)wew·λ(t) hΦ , [Q ]i = (−1)dim(AP ) w∈WK . t λ M◦ P ∆T (t) We refer the readers to Theorem 5.4 for the notations involved the above formulas.
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