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

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.

AMS-TeX, 9 pages