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

Abstract

Let ${\mathfrak g}$ be a simple Lie algebra. For a level $κ$ (thought of as a symmetric ${\mathfrak g}$-invariant form of ${\mathfrak g}$), let $\hat{\mathfrak g}_κ$ be the corresponding affine Kac-Moody algebra. Let $Gr_G$ be the affine Grassmannian of ${\mathfrak g}$, and let $D_κ(Gr_G)-mod$ be the category of $κ$-twisted right D-modules on $Gr_G$. By taking global sections of a D-module, we obtain a functor $Γ:D_κ(Gr_G)-mod\to {\mathfrak g}_κ-mod$. 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.