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

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

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Received 4 June 2021

Published 7 December 2021