The Higson–Mackey analogy for finite extensions of complex semisimple groups

Abstract

In the 1970s, George Mackey pointed out an analogy that exists between tempered representations of semisimple Lie groups and unitary representations of associated semidirect product Lie groups. More recently, Nigel Higson refined Mackey’s analogy into a one-to-one correspondence for connected complex semisimple groups, and in doing so obtained a novel verification of the Baum–Connes conjecture with trivial coefficients for such groups. Here we extend Higson’s results to any Lie group with finitely many connected components whose connected component of the identity is complex semisimple. Our methods include Mackey’s description of unitary representations of group extensions involving projective unitary representations, as well as the notion of twisted crossed product $C^{\ast }$-algebra introduced independently by Green and Dang Ngoc.