Vol. 4, No. 2, 2019

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Orbital integrals and $K\mkern-2mu$-theory classes

Peter Hochs and Hang Wang

Vol. 4 (2019), No. 2, 185–209
Abstract

Let G be a semisimple Lie group with discrete series. We use maps K0(CrG) 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 K0(CrG), 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 K0(CrG) , 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.

Keywords
$K\mkern-2mu$-theory of group $C^*$-algebras, orbital integral, equivariant index, semisimple Lie group, Connes–Kasparov conjecture
Mathematical Subject Classification 2010
Primary: 19K56
Secondary: 22E46, 46L80, 58J20
Milestones
Received: 19 March 2018
Revised: 20 November 2018
Accepted: 6 December 2018
Published: 16 June 2019
Authors
Peter Hochs
School of Mathematical Sciences
University of Adelaide
Adelaide, SA
Australia
Hang Wang
Research Center for Operator Algebras
East China Normal University
Shanghai
China