Non-Wiener groups with a Gelfand pair

TL;DR

This paper investigates the harmonic analysis properties of certain non-amenable groups with Gelfand pairs, showing that under specific boundary representation conditions, these groups do not satisfy Wiener's theorem, with applications to automorphism groups and algebraic groups.

Contribution

It establishes conditions under which non-amenable groups with Gelfand pairs are not Wiener groups, extending understanding of their harmonic analysis properties.

Findings

Groups with certain boundary representations are not Wiener groups.

Automorphism groups of graphs with infinitely many ends are not Wiener.

Split reductive algebraic groups over non-archimedean fields are not Wiener.

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.