Orbital integrals and $K$-theory classes

Abstract

Let $G$ be a semisimple Lie group with discrete series. We use maps $K_0(C_r^{\ast }G)\to \mathbb{C}$ 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 $K_0(C_r^{\ast }G)$, 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 $K_0(C_r^{\ast }G)\to \mathbb{C}$, 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.

arXiv comment29 pages

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Cited by in corpus (3)

• Higher Orbit Integrals, Cyclic Cocyles, and K-theory of Reduced Group C*-algebra
• An index theorem for higher orbital integrals
• An equivariant orbifold index for proper actions