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Volume growth, temperedness and integrability of matrix coefficients on a real spherical space

Friedrich Knop, Bernhard Krötz, Eitan Sayag, Henrik Schlichtkrull

Abstract

We apply the local structure theorem and the polar decomposition to a real spherical space Z=G/H and control the volume growth on Z. We define the Harish-Chandra Schwartz space on Z. We give a geometric criterion to ensure $L^p$-integrability of matrix coefficients on Z.

Volume growth, temperedness and integrability of matrix coefficients on a real spherical space

Abstract

We apply the local structure theorem and the polar decomposition to a real spherical space Z=G/H and control the volume growth on Z. We define the Harish-Chandra Schwartz space on Z. We give a geometric criterion to ensure -integrability of matrix coefficients on Z.

Paper Structure

This paper contains 13 sections, 16 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Homogeneous real spherical spaces
  3. The compression cone and the polar decomposition
  4. The limiting subalgebra
  5. Quasiaffine real spherical spaces
  6. Weight functions
  7. Volume growth
  8. Harish-Chandra Schwartz space on $Z$
  9. $Z$-tempered Harish-Chandra modules
  10. Bounds for generalized matrix coefficients
  11. Property (I)
  12. Factorization
  13. Main theorem on property (I)

Key Result

Theorem 1.1

Let $Z=G/H$ be a wavefront real spherical space with $H$ self-normalizing and reductive. Then $Z$ has property (I).

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Proposition 3.4
  • proof
  • ...and 35 more