The Mackey analogy

If G is a real reductive group, a classical construction in physics is to deform it to its Cartan motion group G_0. In the early 1970's, Mackey conjectured that this procedure produced a bijection between the tempered spectra of G and G_0. The goal of this talk is to introduce some key ideas in harmonic analysis following the work of Harish-Chandra, and apply them to the simple example of SL2(C) to see Mackey's analogy appear. We will then explain how the bijection can be realized by contracting representations of G to G_0.