Analysis Seminar: Mackey Analogy and Orbital Integral

Abstract: The Mackey analogy is a phenomenon in representation theory of Lie groups. If we let G be a (semisimple) Lie group and let G0 be its associated Cartan motion group (a semidirect product of a compact subgroup and an abelian group), then we can find a bijection between equivalence classes of certain unitary representation of G and equivalence classes of unitary representation of G0. In an effort to better understand this analogy, we construct a smooth fiber bundle over the real line called the deformation space where all fibers but one are G. The fiber at zero is G0. In this talk, I will use the Mackey analogy and the deformation space to compare the orbital integrals on G and the orbital integral on G0. First, I will show a result by Yanli Song and Xiang Tang, where they considered the "equal rank" case. In an ongoing joint work with Yanli Song and Xiang Tang, we attempt to extend this result to a more general case.

Host: Walton Green