Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Harish-Chandra's Plancherel theorem for Riemannian symmetric spaces

Bernhard Krötz, Job J. Kuit, Henrik Schlichtkrull

Abstract

In this article we give an overview of the Plancherel theory for Riemannian symmetric spaces Z = G/K. In particular we illustrate recently developed methods in Plancherel theory for real spherical spaces by explicating them for Riemannian symmetric spaces, and we explain how Harish-Chandra's Plancherel theorem for Z can be proven from these methods.

On Harish-Chandra's Plancherel theorem for Riemannian symmetric spaces

Abstract

In this article we give an overview of the Plancherel theory for Riemannian symmetric spaces Z = G/K. In particular we illustrate recently developed methods in Plancherel theory for real spherical spaces by explicating them for Riemannian symmetric spaces, and we explain how Harish-Chandra's Plancherel theorem for Z can be proven from these methods.
Paper Structure (24 sections, 18 theorems, 225 equations)

This paper contains 24 sections, 18 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Notions and Generalities
  3. Normalization of measures and integral formulas
  4. Spherical representation theory
  5. Spherical representations and spherical functions
  6. Abstract Plancherel Theorem for $L^2(Z)$
  7. Irreducible spherical representations
  8. Harish-Chandra's spherical functions
  9. $Z$-tempered representations
  10. Intertwiners and asymptotics
  11. Heuristic from finite dimensional representations
  12. Intertwining operators
  13. Principal asymptotics of $K$-fixed vectors
  14. $Z$-Temperedness revisited
  15. The Plancherel formula for $L^2(Z_\varnothing)$
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $\Phi:G\to \mathbb{C}$ be a continuous function satisfying Then the restriction of $\Phi$ to $N$ is integrable and

Theorems & Definitions (31)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Corollary 3.3
  • proof
  • Proposition 3.4
  • Proposition 3.5
  • Theorem 3.6
  • ...and 21 more