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:
- All irreducible representations of G(𝔽q) can be constructed from a finite set of building blocks — called ‘unipotent representations.’
- Unipotent representations can be classified by certain geometric parameters related to nilpotent orbits for a complex group associated to G(𝔽q).
Now, replace 𝔽q by 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 Losev and Dmytro Matvieievskyi.
This video is part of the University of Georgia‘s Algebra seminar.
