  We outline a proof of a geometric version of the Satake isomorphism. Given a
connected, complex algebraic reductive group G we show that the tensor category
of representations of the dual group $\check G$ is naturally equivalent to a
certain category of perverse sheaves on the loop Grassmannian of G. The above
result has been announced by Ginsburg in \cite{G} and some of the arguments are
borrowed from \cite{G}. However, we use a more "natural" commutativity
constraint for the convolution product, due to Drinfeld. Secondly, we give a
direct geometric proof that the global cohomology functor is exact and
decompose this cohomology functor into a direct sum of weights. We avoid the
use of the decomposition theorem of \cite{BBD} which makes our techniques
applicable to perverse sheaves with coefficients over arbitrary commutative
rings. This note includes sketches of (some of the) proofs. The details, as
well as the generalization to representations over arbitrary fields and
commutative rings will appear elsewhere.
