Abstract: A fundamental problem in representation theory is to determine the unitary dual of a given Lie group G, namely the set of equivalent classes of irreducible unitary representations of G. A principal idea, originated in a famous paper of A. A. Kirillov in 1962, is that there is a close connection between irreducible unitary representations of G and the orbits of G on the dual of its Lie algebra. This is known as the orbit method (or the philosophy of coadjoint orbits).

In this talk, I will describe basic ideas of the orbit method as well as a recent development on the problem of unipotent representations, which is to associate unitary representations to nilpotent coadjoint orbits and which is the hardest part of the orbit method. We solve this problem for real classical groups, by profitably combining analytic ideas of R. Howe on theta lifting and algebro-geometric ideas of D. A. Vogan, Jr. on associate varieties. This is joint work with J.-J. Ma and B. Sun.

The talk is aimed at a general audience of mathematicians and graduate students.