Annals of K-Theory

Orbital integrals and $K$-theory classes

Peter Hochs and Hang Wang

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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.

Article information

Source
Ann. K-Theory, Volume 4, Number 2 (2019), 185-209.

Dates
Received: 19 March 2018
Revised: 20 November 2018
Accepted: 6 December 2018
First available in Project Euclid: 13 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.akt/1565661791

Digital Object Identifier
doi:10.2140/akt.2019.4.185

Mathematical Reviews number (MathSciNet)
MR3990784

Zentralblatt MATH identifier
07102032

Subjects
Primary: 19K56: Index theory [See also 58J20, 58J22]
Secondary: 22E46: Semisimple Lie groups and their representations 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 58J20: Index theory and related fixed point theorems [See also 19K56, 46L80]

Keywords
$K\mkern-2mu$-theory of group $C^*$-algebras orbital integral equivariant index semisimple Lie group Connes–Kasparov conjecture

Citation

Hochs, Peter; Wang, Hang. Orbital integrals and $K$-theory classes. Ann. K-Theory 4 (2019), no. 2, 185--209. doi:10.2140/akt.2019.4.185. https://projecteuclid.org/euclid.akt/1565661791


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