#### 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 ${K}_{0}\left({C}_{r}^{\ast }G\right)\to ℂ$ 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 ${K}_{0}\left({C}_{r}^{\ast }G\right)$, 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 ${K}_{0}\left({C}_{r}^{\ast }G\right)\to ℂ$, 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.

##### 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