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Harish-Chandra's admissibility theorem and beyond

Toshiyuki Kobayashi

TL;DR

The article surveys Harish-Chandra's admissibility for restricting representations to maximal compact subgroups and extends these ideas to non-compact subgroups. It develops two general frameworks—$G'$-admissible restriction (discrete, finite multiplicity) and finite/uniform multiplicity properties—to study branching beyond Riemannian settings, with a focus on real spherical spaces and complexifications. The work provides classification criteria, examples, and methods (D-modules, visible actions) that connect branching laws to spectral theory on standard locally homogeneous spaces, including pseudo-Riemannian locally symmetric spaces. It culminates in a geometry-driven approach to spectral analysis on quotients $\Gamma\backslash G/H$, establishing discretely decomposable restrictions with uniformly bounded multiplicities and a subrepresentation-theoretic backbone for understanding the spectrum in non-classical contexts.

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.

Harish-Chandra's admissibility theorem and beyond

TL;DR

The article surveys Harish-Chandra's admissibility for restricting representations to maximal compact subgroups and extends these ideas to non-compact subgroups. It develops two general frameworks—-admissible restriction (discrete, finite multiplicity) and finite/uniform multiplicity properties—to study branching beyond Riemannian settings, with a focus on real spherical spaces and complexifications. The work provides classification criteria, examples, and methods (D-modules, visible actions) that connect branching laws to spectral theory on standard locally homogeneous spaces, including pseudo-Riemannian locally symmetric spaces. It culminates in a geometry-driven approach to spectral analysis on quotients , establishing discretely decomposable restrictions with uniformly bounded multiplicities and a subrepresentation-theoretic backbone for understanding the spectrum in non-classical contexts.

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.
Paper Structure (38 sections, 15 theorems, 39 equations)

This paper contains 38 sections, 15 theorems, 39 equations.

Table of Contents

  1. Admissible Restriction à la Harish-Chandra
  2. Reminder: Harish-Chandra's admissibility theorem
  3. From Riemannian to reductive symmetric pairs
  4. Admissible Restriction to Non-Compact Subgroups
  5. Admissible restriction $G \downarrow G'$ for a non-compact subgroup
  6. Restriction to compact subgroups $K'$ $(\subset K)$
  7. Admissible restriction $G \downarrow {G'}$
  8. Examples of admissible restrictions $\Pi|_{G'}$
  9. Classification theory: Admissible restriction $\Pi|_{G'}$
  10. Bounded/Finite Multiplicity Pairs for Restriction
  11. Reminder: smooth representations of moderate growth
  12. Multiplicity of the restriction $\Pi|_{G'}$
  13. Comparison: $GL(n,{\mathbb{R}}) \downarrow O(n)$ vs $GL(n,{\mathbb{R}}) \downarrow O(p,n-p)$
  14. Spherical vs real spherical
  15. Analytic view of real spherical spaces
  16. ...and 23 more sections

Key Result

Theorem 1.1

For any irreducible unitary representation $\Pi$ of $G$, one has the following finite-multiplicity property: Here $[\Pi|_K: \pi]$ denotes the multiplicity of the representation $\pi$ occurring in the restriction of $\Pi$ to $K$.

Theorems & Definitions (44)

  • Theorem 1.1: Harish-Chandra's admissibility theorem
  • Definition 1.2
  • Example 1.3
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Criterion for $K'$-admissibility
  • Corollary 2.4: Criterion for admissible restriction
  • Remark 2.5
  • Example 2.6: Theta correspondence
  • Example 2.7
  • ...and 34 more