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 G0 − 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 G0. 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.