A geometric realisation of tempered representations restricted to maximal compact subgroups

TL;DR

This paper provides a geometric realization of the restriction of tempered representations of real reductive Lie groups to maximal compact subgroups, using Dirac operators on homogeneous spaces linked to coadjoint orbits, extending Kirillov's orbit method.

Contribution

It introduces a new geometric model for the restriction of tempered representations to maximal compact subgroups, generalizing previous results to all tempered representations.

Findings

Realizes $ ext{pi}|_K$ as a $K$-equivariant index of a Dirac operator.

Identifies the space with a coadjoint orbit, providing an explicit orbit method.

Prepares for a geometric formula for $K$-type multiplicities in a companion paper.

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.