Special unipotent representations of classical Lie groups

One fundamental problem in representation theory is the unitary dual problem, namely to construct and classify all irreducible unitary representations of a given Lie group $G$. An important principle is the orbit method introduced by A. A. Kirillov, and it seeks to describe irreducible unitary representations of $G$ by its coadjoint orbits. The most mysterious ingredient of orbit method is to attach irreducible unitary representations to nilpotent coadjoint orbits. For classical Lie groups, we construct some irreducible unitary representations attached to nilpotent coadjoint orbits, by using the theory of local theta correspondence initiated by R. Howe. These are the special unipotent representations in the sense of Arthur and Barbasch-Vogan. This is a report on a recent joint work with Dan M. Barbarsch, Jia-Jun Ma and Chen-Bo Zhu.