David Vogan
Massachusetts Institute of Technology
Nilpotent elements and reductive groups

In the study of conjugacy classes of complex n x n matrices, nilpotent matrices play a central role. First, there are just finitely many nilpotent conjugacy classes: one for each partition of n. It is therefore possible to study and understand each class individually.

Second, any conjugacy class of n x n matrices looks approximately like a nilpotent conjugacy class. (In the language of algebraic geometry the closure of any conjugacy class is a deformation of the closure of a unique nilpotent conjugacy class.) Third, there is a close connection between nilpotent conjugacy classes and (infinite-dimensional irreducible representations.

All of these assertions have analogues with invertible matrices replaced by any reductive algebraic group G. It is therefore of great interest to understand the analogue of "nilpotent conjugacy classes" for such a group, to classify these, and to understand each one. Although the classification problem was solved by Dynkin almost sixty years ago, we still do not understand the answer well.

I will describe Dynkin's (case by case) solution to the classification problem, and talk about what a better solution might look like. I will then discuss how to attach a nilpotent orbit to each irreducible representation; (tools to compute this correspondence explicitly problem that will probably be considered more thoroughly in Binegar's lectures); and how to use the correspondence to study unitary representations.