A program for branching problems in the representation theory of real reductive groups

TL;DR

This paper explores the branching problem in the representation theory of real reductive groups, focusing on understanding how irreducible representations decompose when restricted to subgroups, and introduces new perspectives and methods for analysis.

Contribution

It provides new insights and methods for analyzing branching laws and symmetry breaking operators in the context of real reductive groups, highlighting recent progress and open questions.

Findings

Detailed analysis of branching laws in specific settings

Introduction of new methods for constructing symmetry breaking operators

Open questions and conjectures for future research

Abstract

We wish to understand how irreducible representations of a group G behave when restricted to a subgroup G' (the branching problem). Our primary concern is with representations of reductive Lie groups, which involve both algebraic and analytic approaches. We divide branching problems into three stages: (A) abstract features of the restriction; (B) branching laws (irreducible decompositions of the restriction); and (C) construction of symmetry breaking operators on geometric models. We could expect a simple and detailed study of branching problems in Stages B and C in the settings that are {\it{a priori}} known to be "nice" in Stage A, and conversely, new results and methods in Stage C that might open another fruitful direction of branching problems including Stage A. The aim of this article is to give new perspectives on the subjects, to explain the methods based on some recent progress, and to raise some conjectures and open questions.