Branching laws and a duality principle, Part I

TL;DR

This paper investigates how Discrete Series representations of semisimple Lie groups decompose when restricted to subgroups, using integral and differential operators alongside a duality principle, with detailed focus on Holomorphic Discrete Series.

Contribution

It extends the understanding of branching laws for Discrete Series representations using new analytical methods and duality principles, providing both general results and detailed cases.

Findings

Derived new branching laws for Discrete Series

Applied duality principles to representation restriction

Provided detailed analysis for Holomorphic Discrete Series

Abstract

For a semisimple Lie group $G$ satisfying the equal rank condition, the most basic family of unitary irreducible representations is the Discrete Series found by Harish-Chandra. In this paper, we continue our study of the branching laws for Discrete Series when restricted to a subgroup $H$ of the same type by use of integral and differential operators in combination with our previous duality principle. Many results are presented in generality, others are shown in detail for Holomorphic Discrete Series.