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2019 Orbital integrals and $K$-theory classes
Peter Hochs, Hang Wang
Ann. K-Theory 4(2): 185-209 (2019). DOI: 10.2140/akt.2019.4.185

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.

Citation

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Peter Hochs. Hang Wang. "Orbital integrals and $K$-theory classes." Ann. K-Theory 4 (2) 185 - 209, 2019. https://doi.org/10.2140/akt.2019.4.185

Information

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

zbMATH: 07102032
MathSciNet: MR3990784
Digital Object Identifier: 10.2140/akt.2019.4.185

Subjects:
Primary: 19K56
Secondary: 22E46, 46L80, 58J20

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.4 • No. 2 • 2019
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