Let G be a connected semisimple Lie group with finite center, and let K be a maximal compact subgroup. Mackey suggested that there should be an analogy between almost all the unitary representations of G and almost all of the unitary representations of an associated semidirect product group G₀ = K [special characters omitted], where [special characters omitted] and [special characters omitted] are the Lie algebras of G and K, respectively.

Perhaps because Mackey typically viewed the dual of a Lie group as a Borel space, the analogy he proposed was between almost all unitary representations of G and G₀. In the case of complex semisimple Lie groups, Higson completed Mackey's analogy at the level of parameters, constructing a reasonably natural bijection between all irreducible tempered representations of G and all irreducible unitary representations of G₀. In this dissertation, following Higson, and relying heavily on Vogan's classification of representations by minimal K-types, we conjecture that there is a very natural bijection between Ĝ and Ĝ₀ for any connected semisimple Lie group with finite center, and prove this conjecture for G = SL(n, [special characters omitted]).