Harish-Chandra's admissibility theorem and beyond

TL;DR

This paper reviews Harish-Chandra's admissibility theorem and explores recent generalizations to non-compact subgroups, highlighting new spectral analysis methods for locally symmetric spaces beyond classical settings.

Contribution

It introduces two new directions for extending Harish-Chandra's work to non-compact subgroups, enhancing spectral analysis techniques.

Findings

Development of discrete decomposability with finite multiplicity

Finite/uniformly bounded multiplicity properties

Application to spectral analysis of locally symmetric spaces

Abstract

This article is a record of the lecture at the centennial conference for Harish-Chandra. The admissibility theorem of Harish-Chandra concerns the restrictions of irreducible representations to maximal compact subgroups. In this article, we begin with a brief explanation of two directions for generalizing his pioneering work to {\it{non-compact}} reductive subgroups: one emphasizes discrete decomposability with the finite multiplicity property, while the other focuses on finite/uniformly bounded multiplicity properties. We discuss how the recent representation-theoretic developments in these directions collectively offer a powerful method for the new spectral analysis of standard locally symmetric spaces, extending beyond the classical Riemannian setting.