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

TL;DR

This paper provides a geometric formula for the multiplicities of $K$-types in tempered representations of real reductive Lie groups, extending previous results and offering criteria for multiplicity-free restrictions.

Contribution

It introduces a new geometric expression for $K$-type multiplicities in tempered representations, generalizing earlier work and applying the orbit method and quantization principles.

Findings

Derived a geometric formula for $K$-type multiplicities.

Established criteria for multiplicity-free restrictions.

Applied results to specific groups like $ ext{SU}(p,1)$, $ ext{SO}_0(p,1)$, and $ ext{SO}_0(2,2)$.

Abstract

Let $G$ be a connected, linear, real reductive Lie group with compact centre. Let $K<G$ be compact. Under a condition on $K$, which holds in particular if $K$ is maximal compact, we give a geometric expression for the multiplicities of the $K$-types of any tempered representation (in fact, any standard representation) $\pi$ of $G$. This expression is in the spirit of Kirillov's orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of $\pi|_K$ obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of $\mathrm{SU}(p,1)$, $\mathrm{SO}_0(p,1)$ and $\mathrm{SO}_0(2,2)$ restrict multiplicity-freely to maximal compact subgroups.