On the analogy between real reductive groups and Cartan motion groups. I: The Mackey-Higson bijection

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.

Version 5 (16 pages), to appear in the Cambridge Journal of Math

Cited by in corpus (4)

• Orbital integrals and $K$-theory classes
• Algebraic Families of Groups and Commuting Involutions
• On the analogy between real reductive groups and Cartan motion groups. III: A proof of the Connes-Kasparov isomorphism
• Continuity of the Mackey-Higson bijection