## Annals of K-Theory

### Orbital integrals and $K$-theory classes

#### Abstract

Let $G$ be a semisimple Lie group with discrete series. We use maps $K0(Cr∗G)→ℂ$ 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 $K0(Cr∗G)$, 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 $K0(Cr∗G)→ℂ$, 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.

#### Article information

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

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

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

#### 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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