  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.
