The Plancherel formula for countable groups

TL;DR

This paper develops a Plancherel formula for countable groups, decomposing their regular representation into factor representations, and applies it to linear groups, providing insights into their harmonic analysis and representation types.

Contribution

It offers a new description of the Plancherel decomposition for countable groups using the FC-center and automorphism actions, extending harmonic analysis tools.

Findings

Provides a detailed Plancherel formula for countable groups.

Determines the Plancherel formula specifically for linear groups.

Offers a unified proof of Thoma's and Kaniuth's theorems regarding group types.

Abstract

We discuss a Plancherel formula for countable groups, which provides a canonical decomposition of the regular representation of such a group $\Gamma$ into a direct integral of factor representations. Our main result gives a precise description of this decomposition in terms of the Plancherel formula of the FC-center $\Gamma_{\rm fc}$ of $\Gamma$ (that is, the normal sugbroup of $\Gamma$ consisting of elements with a finite conjugacy class); this description involves the action of an appropriate totally disconnected compact group of automorphisms of $\Gamma_{\rm fc}$. As an application, we determine the Plancherel formula for linear groups. In an appendix, we use the Plancherel formula to provide a unified proof for Thoma's and Kaniuth's theorems which respectively characterize countable groups which are of type I and those whose regular representation is of type II.