  In the 1970's, 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 $C^*$-algebra introduced independently by Green and
Dang Ngoc.
