The Mackey bijection as a stratified equivalence

TL;DR

This paper investigates the Mackey bijection's behavior on stratifications of tempered duals and reduced C*-algebras, revealing new topological insights and formulating stratified equivalence inspired by p-adic group studies.

Contribution

It introduces a stratified equivalence framework for the Mackey bijection and analyzes its topological properties within the context of reduced C*-algebras and tempered dual stratifications.

Findings

Behavior of the Mackey embedding on stratified reduced C*-algebras analyzed.

New topological properties of the Mackey bijection derived.

Formulation of stratified equivalence inspired by p-adic group structures.

Abstract

This paper is about the Mackey analogy between the tempered representation theory of a real reductive group and that of its Cartan motion group. We consider the embedding of reduced C*-algebras constructed recently in connection with the Mackey bijection, and study its behavior on certain natural stratifications of the tempered duals. We formulate our result using a notion of stratified equivalence inspired by the study of the smooth dual of $p$-adic groups via the structure of Hecke algebras, in particular by the work of Aubert, Baum, Plymen and Solleveld. We derive related new topological properties of the Mackey bijection. We also analyze the behavior of the Mackey embedding on a stratification of reduced C*-algebras attached to a partition of the tempered dual into particularly elementary pieces, introduced in recent work of Bradd, Higson and Yuncken.