Cambridge Journal of Mathematics

Volume 9 (2021)

Number 3

On the analogy between real reductive groups and Cartan motion groups: the Mackey–Higson bijection

Pages: 551 – 575

DOI:  https://dx.doi.org/10.4310/CJM.2021.v9.n3.a1

Author

Alexandre Afgoustidis (CNRS, Institut Élie Cartan de Lorraine, Université de Lorraine, Nancy & Metz, France)

Abstract

George Mackey suggested in 1975 that there should be analogies between the irreducible unitary representations of a noncompact reductive Lie group $G$ and those of its Cartan motion group $G_0$ — the semidirect product of a maximal compact subgroup of $G$ and a vector space. He conjectured the existence of a natural one-to-one correspondence between “most” irreducible (tempered) representations of $G$ and “most” irreducible (unitary) representations of $G_0$. We here describe a simple and natural bijection between the tempered duals of both groups, and an extension to a one-to-one correspondence between the admissible duals.

Keywords

real reductive groups, Cartan motion group, Mackey analogy

2010 Mathematics Subject Classification

22E46, 22E50

Received 4 June 2021

Published 7 December 2021