Continuity of the Mackey–Higson bijection

Alexandre Afgoustidis and Anne-Marie Aubert

Vol. 310 (2021), No. 2, 257–273

DOI: 10.2140/pjm.2021.310.257

Abstract

When $G$ is a real reductive group and $G_0$ is its Cartan motion group, the Mackey–Higson bijection is a natural one-to-one correspondence between all irreducible tempered representations of $G$ and all irreducible unitary representations of $G_0$. We collect some known facts about the topology of the tempered dual $\tilde{G}$ and that of the unitary dual $\widehat{G_0}$, and then verify that the Mackey–Higson bijection $\tilde{G}\to \widehat{G_0}$ is continuous.

Keywords

real reductive groups, Cartan motion group, tempered representations, Fell topology, Mackey–Higson bijection

Mathematical Subject Classification 2010

Primary: 22E47

Secondary: 22E50

Milestones

Received: 8 November 2019

Revised: 16 November 2020

Accepted: 9 December 2020

Published: 8 March 2021