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 $\hat{\mathfrak{g}}_\kappa$ 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 → $\mathfrak{g}$_κ-mod. It is known that this functor is exact and faithful when κ is less than critical or irrational. In this paper, we show that the functor Γ is also exact and faithful when κ is the critical level.