# Branching problems in reproducing kernel spaces

## TL;DR

This paper investigates how discrete series representations of semisimple Lie groups decompose when restricted to subgroups, linking discrete decomposability to differential operator representations and reproducing kernel conditions.

## Contribution

It establishes new conditions for discrete decomposability of discrete series representations upon restriction, connecting classical and recent results with differential operator frameworks.

## Key findings

- Proves discrete decomposability under Harish-Chandra's cusp form condition
- Links decomposability to differential operator representations
- Provides new branching laws for discrete series representations

## 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 study some of the branching laws for discrete series when restricted to a subgroup $H$ of the same type by combining classical results with recent work of T. Kobayashi; in particular, we prove discrete decomposability under Harish-Chandra's condition of cusp form on the reproducing kernel. We show a relation between discrete decomposability and representing certain intertwining operators in terms of differential operators.