  In this paper we give a geometric version of the Satake isomorphism. Given a
connected complex reductive algebraic group, we show that the category of
representations of its Langlands dual is naturally equivalent to a certain
category of perverse sheaves on the complex affine Grassmannian. We can work
with perverse sheaves with coefficients in an arbitrary commutative ring and
then we recover the representation theory of the split form of the dual group
over the commutative ring.

Representation Theory;Algebraic Geometry