Application of Mackey deformation; reduced group C*-algebra and cyclic cohomology

The Mackey bijection, or the Mackey analogy, is a phenomenon in representation theory that describes a one-to-one correspondence between equivalence classes of irreducible tempered representation of a reductive group, and equivalence classes of irreducible unitary representation of an associated motion group. The deformation to the normal cone, which we shall call the Mackey deformation, is a smooth differentiable manifold that fits the reductive group and the associated motion group as fibers over the real line. In this talk, I shall introduce the Mackey deformation, then explain a few applications. Some applications include constructing an embedding from the group C*-algebra of the motion group into the reduced C*-algebra of the reductive group and a limit formula involving cyclic cohomologies of both the real reductive group and the motion group.  In particular, the so-called higher orbital integrals are generators of the cyclic cohomologies; I shall introduce both the higher orbital integrals and the cyclic cohomology.

Time permitting, I will also describe current and future research direction involving the Mackey deformation. Many of the results that will be presented are joint work with many collaborators that I shall mention throughout the talk.