D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras

Let g be a simple Lie algebra. For a level kappa (thought of as a symmetric g-invariant form of g), let g(kappa) be the corresponding affine Kac-Moody algebra. Let Gr(G) be the affine Grassmannian of g, and let D-kappa(Gr(G))-mod be the category Oft-twisted right D-modules on GrG. By taking global sections of a D-module we obtain a functor Gamma : D-kappa(GrG)-mod --> g(kappa)-mod. It is known that this functor is exact and faithful when kappa is less than critical or irrational. In this paper, we show that the functor Gamma is also exact and faithful when kappa is the critical level.