Orbit Method: From Matrices to Unitary Representations
Abstract: The talk is intended as a leisurely introduction to one of the fundamental tasks of representation theory: the construction of irreducible unitary representations. I will start with the basic notions in representation theory and then discuss two philosophical sources of unitary representations of Lie groups: symplectic geometry (Kirillov theory) and harmonic analysis (Arthur’s classification). In the second part of the talk, I will discuss a recent development (due to Barbasch, Ma, Sun and the speaker) on Vogan’s problem of unipotent representations, which is to associate irreducible unitary representations to nilpotent matrices (which behave like zero but are not zero). I will explain how we solve the mystery for those nilpotent matrices which are special in the sense of Lusztig.