Kirillov's orbit method: the case of discrete series representations

TL;DR

This paper provides a geometric formula for calculating G-multiplicities in discrete series representations of real semi-simple Lie groups, focusing on cases where the representation is G-admissible.

Contribution

It introduces a new geometric approach to determine G-multiplicities in discrete series representations, extending Kirillov's orbit method to this context.

Findings

Derived explicit geometric expressions for G-multiplicities

Extended Kirillov's orbit method to discrete series representations

Provided conditions for G-admissibility in representations

Abstract

Let V be an Harish-Chandra discrete series representation of a real semi-simple Lie group G' and let G be a semi-simple subgroup of G'. In this paper, we give a geometric expression of the G-multiplicities in V when the representation V is supposed to be G-admissible.