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

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