Recent advances in branching problems of representations

TL;DR

This paper reviews recent developments in the study of how irreducible representations of groups behave when restricted to subgroups, emphasizing new directions in the representation theory of real reductive groups and their applications.

Contribution

It provides an up-to-date overview of recent advances in branching problems for real reductive groups, connecting representation theory with global analysis and discontinuous groups.

Findings

New directions in branching problems for real reductive groups

Connections between representation theory and global analysis of manifolds

Insights into discontinuous groups beyond classical Riemannian settings

Abstract

How does an irreducible representation of a group behave when restricted to a subgroup? This is part of branching problems, which are one of the fundamental problems in representation theory, and also interact naturally with other fields of mathematics. This expository paper is an up-to-date account on some new directions in representation theory highlighting the branching problems for real reductive groups and their related topics ranging from global analysis of manifolds via group actions to the theory of discontinuous groups beyond the classical Riemannian setting. This article is an outgrowth of the invited lecture that the author delivered at the commemorative event for the 70th anniversary of the re-establishment of the Mathematical Society of Japan, and originally appeared in Japanese in Sugaku 71 (2019).