On the action of the dual group on the cohomology of perverse sheaves on the affine grassmannian

TL;DR

This paper constructs an explicit action of the dual group on the global cohomology of perverse sheaves on the affine Grassmannian, building on the equivalence of categories established by Ginzburg and Mirkovic-Vilonen.

Contribution

It provides an explicit construction of the dual group action on cohomology, complementing the existing Tannakian formalism proof.

Findings

Explicit dual group action on cohomology constructed

Enhances understanding of the geometric Satake correspondence

Connects perverse sheaves with dual group representations

Abstract

It was proved by Ginzburg and Mirkovic-Vilonen that the $G(O)$-equivariant perverse sheaves on the affine grassmannian of a connected reductive group $G$ form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group $G^\vee$. The proof use the Tannakian formalism. The purpose of this paper is to construct explicitely the action of $G^\vee$ on the global cohomology of a perverse sheaf.