What is a unipotent representation?

Abstract: The concept of a unipotent representation has its origins in the representation theory of finite Chevalley groups. Let G(Fq) be the group of Fq-rational points of a connected reductive algebraic group G. In 1984, Lusztig completed the classification of irreducible representations of G(Fq). He showed:

1) All irreducible representations of G(Fq) can be constructed from a finite set of building blocks -- called `unipotent representations.'

2) Unipotent representations can be classified by certain geometric parameters related to nilpotent orbits for a complex group associated to G(Fq).

Now, replace Fq with C, the field of complex numbers, and replace G(Fq) with G(C). There is a striking analogy between the finite-dimensional representation theory of G(Fq) and the unitary representation theory of G(C). This analogy suggests that all unitary representations of G(C) can be constructed from a finite set of building blocks -- called `unipotent representations' -- and that these building blocks are classified by geometric parameters related to nilpotent orbits. In this talk I will propose a definition of unipotent representations, generalizing the Barbasch-Vogan notion of `special unipotent'. The definition I propose is geometric and case-free. After giving some examples, I will state a geometric classification of unipotent representations, generalizing the well-known result of Barbasch-Vogan for special unipotents.

This talk is based on forthcoming joint work with Ivan Loseu and Dmitryo Matvieievskyi.

mathematical physicsalgebraic geometrycategory theoryrepresentation theory

Audience: researchers in the topic

UMass Amherst Representation theory seminar