Orbital integrals and K-theory classes

Abstract

Let G be a semisimple Lie group with discrete series. We use maps K 0 (C * r G) → C defined by orbital integrals to recover group theoretic information about G, including information contained in K-theory classes not associated to the discrete series. An important tool is a fixed point formula for equivariant indices obtained by the authors in an earlier paper. Applications include a tool to distinguish classes in K 0 (C * r G), the (known) injectivity of Dirac induction, versions of Selberg's principle in K-theory and for matrix coefficients of the discrete series, a Tannaka-type duality, and a way to extract characters of representations from K-theory. Finally, we obtain a continuity property near the identity element of G of families of maps K 0 (C * r G) → C, parametrised by semisimple elements of G, defined by stable orbital integrals. This implies a continuity property for L-packets of discrete series characters, which in turn can be used to deduce a (well-known) expression for formal degrees of discrete series representations from Harish-Chandra's character formula.

An equivariant orbifold index for proper actions

“…In several contexts [10,16,27], it was shown that traces defined by orbital integrals on discrete groups are useful tools to extract information from classes in K-theory of group C * -algebras. (This is also true for orbital integrals on semisimple Lie groups [13] and higher analogues [25].) For a discrete group Γ, the orbital integral of a function f ∈ l 1 (Γ) over a conjugacy class (γ) of an element γ ∈ Γ, is the sum of f over (γ):…”

An equivariant Atiyah–Patodi–Singer index theorem for proper actions II: the K-theoretic index

“…It has been applied to various problems in geometry and topology, such as questions about positive scalar curvature and the Novikov conjecture. In this context, the number (1.5) was shown to be relevant to representation theory, orbifold geometry and trace formulas [22,24,25,40]. • If M and G are compact, then (1.2) becomes the equivariant APS index used in [14], and…”

Higher Orbit Integrals, Cyclic Cocyles, and K-theory of Reduced Group C*-algebra

“…where C ∞ (H reg ) −W(H,G) is the space of anti-symmetric functions with respect to the Weyl group W(H, G) action on H, ǫ H (h) is a sign function on H, and ∆ G H is the Weyl denominator for H. Our starting point is the following property that for a given h ∈ H reg , the linear functional on S(G), F H (h) : f → F H f (h), is a trace on S(G), c.f. [11]. In cyclic cohomology, traces are special examples of cyclic cocycles on an algebra.…”

“…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 representation ring of K, a maximal compact subgroup of G. Nevertheless, when G has non-equal rank, F H vanishes on K-theory of S(G) completely, c.f. [11]. Furthermore, many numerical invariants for G-equivariant Dirac operators in literature, e.g .…”

“…Furthermore, many numerical 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.…”