报告人：Jing-Song Huang (The Hong Kong University of Science and Technology)
地点：Room 1556(1560), Sciences Building No. 1
Abstract: Classifying irreducible unitary representations of real reductive Lie groups is a central problem
in representation theory, which is well-known as the unitary dual problem. The orbit method establishes a
correspondence between irreducible unitary representations of a Lie group and its coadjoint orbits. Diximer's
work on primitive ideals is one of the earliest indications of such a correspondence. The theory was established
by Kirillov for nilpotent groups and it was later extended by Bertram Kostant and Louis Auslander to solvable
groups. David Vogan proposed that the orbit method should serve as a unifying principle in the description of the unitary dual of real reductive groups.
In Vogan's formulation of the orbit method for real reductive groups, the correspondence from the coadjoint
orbits to irreducible unitary representations is divided into three steps according to the Jordan decomposition
of a linear functional on Lie algebras into hyperbolic, ellipticand nilpotent components. The hyperbolic step
and elliptic step are well understood, while the nilpotent step to construct unipotent representations from
nilpotent orbits has been extensively studied from several different perspectives over the last thirty years.
Nevertheless, the final definition of unipotent representations remains a mystery.
The aim of this talk is to show that our recent work joint with Pandzic and Vogan on Classifying unitary
representations by their Dirac cohomology shed light on what kind of irreducible unitary representations
should be defined as unipotent. Our results suggest that we shall approach the problem by study the coherent continuation of unitary representations with nonzero Dirac cohomology.