  We describe the equivariant cohomology of cofibers of spherical perverse
sheaves on the affine Grassmannian of a reductive algebraic group in terms of
the geometry of the Langlands dual group. In fact we give two equivalent
descriptions: one in terms of D-modules of the basic affine space, and one in
terms of intertwining operators for universal Verma modules. We also construct
natural collections of isomorphisms parametrized by the Weyl group in these
three contexts, and prove that they are compatible with our isomorphisms. As
applications we reprove some results of the first author and of
Braverman-Finkelberg.
