Continuity of the Mackey-Higson bijection

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 $G_0$. In this short note, we collect some known facts about the topology of the tempered dual $\tilde{G}$ and that of the unitary dual $\widehat{G_0}$, then verify that the Mackey-Higson bijection $\tilde{G}\to \widehat{G_0}$ is continuous.

arXiv comment12 pages, to appear in the Pacific Journal of Mathematics

References in corpus (4)

• Parabolic induction and restriction via C*-algebras and Hilbert C*-modules
• On the analogy between real reductive groups and Cartan motion groups. I: The Mackey-Higson bijection
• On the analogy between real reductive groups and Cartan motion groups. II: Contraction of irreducible tempered representations
• On the analogy between real reductive groups and Cartan motion groups. III: A proof of the Connes-Kasparov isomorphism