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.

62 pages. The earlier version of this preprint was split into two; this is the first part

References in corpus (3)

• Applications of Equivariant Cohomology
• Geometric quantization and families of inner products
• The index theory on non-compact manifolds with proper group action

Cited by in corpus (1)

• A geometric formula for multiplicities of $K$-types of tempered representations