Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian and geometric Satake equivalence

Abstract: This talk is based on the paper (joint with M. Finkelberg and I. Mirković).

Let G be a reductive complex algebraic group. Recall that a geometric Satake isomorphism is an equivalence between the category of G(O)-equivariant perverse sheaves on the affine Grassmannian for G and the category of finite dimensional representations of the Langlands dual group \hat{G}. It follows that for any G(O)-equivariant perverse sheaf P there exists an action of the dual Lie algebra \hat{\mathfrak{g}} on the global cohomology of P.

We will explain one possible approach to constructing this action. To do so, we will describe a new geometric construction of the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra \hat{\mathfrak{g}} based on certain one-parametric deformation of zastava spaces. We will introduce the so-called Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian that is a one-parametric deformation of the affine Grassmannian Gr_G. We will discuss the case G=SL_2 as an example.

representation theory

Audience: researchers in the topic

MIT Lie groups seminar

Series comments: Description: Research seminar on Lie groups

This seminar will take place entirely online: Zoom Meeting Link. You should be able to watch live video at this link. Your microphone will be muted, but you are welcome to unmute it (microphone icon on the lower left of the Zoom window, perhaps visible only when you put your mouse near there) to ask a question.