A geometric realisation of tempered representations restricted to maximal compact subgroups

Abstract

Let G be a connected, linear, real reductive Lie group with compact centre. Let $$K<G$$ be maximal compact. For a tempered representation $$\pi $$ of G, we realise the restriction $$\pi |_K$$ as the K-equivariant index of a Dirac operator on a homogeneous space of the form G/H, for a Cartan subgroup $$H<G$$ . (The result in fact applies to every standard representation.) Such a space can be identified with a coadjoint orbit of G, so that we obtain an explicit version of Kirillov’s orbit method for $$\pi |_K$$ . In a companion paper, we use this realisation of $$\pi |_K$$ to give a geometric expression for the multiplicities of the K-types of $$\pi $$ , in the spirit of the quantisation commutes with reduction principle. This generalises work by Paradan for the discrete series to arbitrary tempered representations.