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

Let $\mathfrak{g}$ be a simple Lie algebra. For a level $\kappa$ (thought of as a symmetric $\mathfrak{g}$-invariant form of $\mathfrak{g}$), let ${\widehat{\mathfrak{g}}}_{\kappa }$ be the corresponding affine Kac-Moody algebra. Let $G{r}_G$ be the affine Grassmannian of $\mathfrak{g}$, and let $D_{\kappa }(G{r}_G)-mod$ be the category of $\kappa$-twisted right D-modules on $G{r}_G$. By taking global sections of a D-module, we obtain a functor $\mathrm{\Gamma }:D_{\kappa }(G{r}_G)-mod\to {\mathfrak{g}}_{\kappa }-mod$. It is known that this functor is exact and faithful when $\kappa$ is negative or irrational. In this paper, we show that the functor $\mathrm{\Gamma }$ is exact and faithful also when $\kappa$ is the critical level.