Abstract
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.
Citation
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
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