Annals of K-Theory
- Ann. K-Theory
- Volume 4, Number 2 (2019), 185-209.
Orbital integrals and $K$-theory classes
Let be a semisimple Lie group with discrete series. We use maps defined by orbital integrals to recover group theoretic information about , including information contained in -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 , the (known) injectivity of Dirac induction, versions of Selberg’s principle in -theory and for matrix coefficients of the discrete series, a Tannaka-type duality, and a way to extract characters of representations from -theory. Finally, we obtain a continuity property near the identity element of of families of maps , parametrised by semisimple elements of , defined by stable orbital integrals. This implies a continuity property for -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.
Ann. K-Theory, Volume 4, Number 2 (2019), 185-209.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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