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George Mackey suggested in 1975 that there should be analogies between theirreducible unitary representations of a noncompact reductive Lie group $G$ andthose of its Cartan motion group $G_0$ $-$ the semidirect product of a maximalcompact subgroup of $G$ and a vector space. He conjectured the existence of anatural 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 temperedduals of both groups, and an extension to a one-to-one correspondence betweenthe admissible duals.

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