Topics on global analysis of manifolds and representation theory of reductive groups

TL;DR

This paper reviews recent advances in understanding how geometric symmetries influence the representation theory of reductive groups, particularly affecting multiplicities and spectral properties of function spaces.

Contribution

It summarizes recent theoretical developments connecting geometric conditions to representation theoretic properties in the context of reductive groups.

Findings

Geometric symmetries impact multiplicities in representation spaces.

Spectral properties are influenced by geometric conditions.

Recent theories clarify the relationship between geometry and representation theory.

Abstract

Geometric symmetry induces symmetries of function spaces, and the latter yields a clue to global analysis via representation theory. In this note we summarize recent developments on the general theory about how geometric conditions affect representation theoretic properties on function spaces, with focus on multiplicities and spectrum.