On the Higson-Mackey analogy, group C*-algebras, and K-theory

Let G be a Lie group with finitely many connected components. Let K be a maximal compact subgroup of G. Let G0 = K ⋉g/k be the associated semidirect product Lie group in which g and k denote the respective Lie algebras of G and K, and K acts on their quotient via the adjoint representation. In the 1970's, George Mackey pointed out an analogy between the representation theories of G and G 0 when G is connected and semisimple. Recently, Nigel Higson used operator algebra theory to make the Mackey analogy precise for connected complex semisimple groups. Resulting from Higson's analysis is a new form of verification that such groups satisfy the Baum-Connes conjecture, which aims to describe the K-theory of group C*-algebras.;This dissertation examines the Higson-Mackey analogy beyond connected complex semisimple groups. Our primary focus is the extension of Higson's results to Lie groups with finitely many connected components and complex semisimple identity component. Dealing with these so-called almost connected groups requires some additional concepts from representation theory and operator algebra theory. Nonetheless, we are able to demonstrate that Higson's analysis and consequent verification of the Baum-Connes conjecture can indeed be generalized. Lastly, following an examination of the group SL(2, R ), we describe how our techniques can be applied to a particular almost connected group that introduces some exceptional phenomena into the representation theory of real reductive groups.