Drinfeld-Gaitsgory interpolation Grassmannian and geometric Satake equivalence

開催日時
2018/10/03 水 10:30 - 12:00
場所
3号館108号室
講演者
Vasily Kryov
講演者所属
Higher School of Economics
概要

This talk is a review of the paper arxiv.org/abs/1805.07721. 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 repre- sentations of the Langlands dual group Gˇ. They are equivalent as Tannakian categories, the fiber functor sends a perverse sheaf to its global cohomology. It follows from the above that for any perverse sheaf P there exists an action of the Lie algebra of Gˇ on the global cohomology of P.
We will explain how to construct this action explicitly. To do so, we will describe a geometric construction of the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra U(nˇ). Using this construction, we will provide the desired action. It will be obtained via a cospecialization morphism for a certain one-parametric deformation of the affine Grassmannian of G.

If time permits, we will discuss some possible generalizations of our construction of the action, in particular, we will discuss the relation of the deformation mentioned above with the Drinfeld-Gaitsgory deformation considered in their paper on Braden’s theorem.