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Non-Wiener groups with a Gelfand pair

Max Carter, Jared T. White

Abstract

Let $G$ be a non-amenable locally compact group and $K$ a compact subgroup of $G$ such that $(G,K)$ is a Gelfand pair. We show that if $G$ admits a suitable boundary representation which is topologically irreducible and not unitarizable, then $G$ is not a Wiener group in the sense that its Fourier transform does not satisfy the analogue of Wiener's Tauberian theorem. As an application, we show that if $G$ is a closed non-compact boundary transitive group of automorphisms of a connected locally finite graph with infinitely many ends, or a split reductive algebraic group over a non-archimedean local field, then $G$ is not Wiener.

Non-Wiener groups with a Gelfand pair

Abstract

Let be a non-amenable locally compact group and a compact subgroup of such that is a Gelfand pair. We show that if admits a suitable boundary representation which is topologically irreducible and not unitarizable, then is not a Wiener group in the sense that its Fourier transform does not satisfy the analogue of Wiener's Tauberian theorem. As an application, we show that if is a closed non-compact boundary transitive group of automorphisms of a connected locally finite graph with infinitely many ends, or a split reductive algebraic group over a non-archimedean local field, then is not Wiener.
Paper Structure (14 sections, 19 theorems, 44 equations)

This paper contains 14 sections, 19 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on representations and Gelfand pairs
  3. Conventions
  4. Gelfand pairs
  5. Smooth and admissible representations
  6. Contragradient representations and matrix coefficients
  7. Induced representations
  8. Boundary representations and the Iwasawa decomposition
  9. A theorem of Godement
  10. Proof of Theorem A
  11. Proof of Theorem B
  12. Proof of Theorem B(i)
  13. Proof of Theorem B(ii)
  14. Acknowledgements

Key Result

Proposition 2.3

vD09 Let $G$ be a locally compact group and $K$ a compact subgroup of $G$ such that $(G,K)$ is a Gelfand pair. Let $\varphi: G \rightarrow \mathbb{C}$ be a continuous $K$-bi-invariant function such that $\varphi(\text{\rm id}_G) = 1$. Then the following are equivalent:

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Corollary 2.8
  • proof
  • ...and 28 more