D–modules on the Affine Grassmannian and Representations of Affine Kac-moody Algebras

x⊗ f(t), y ⊗ g(t) 7→ −κ(x, y) · Rest=0 fdg. Denote by ĝκ–mod the category of ĝκ-modules which are discrete, i.e., any vector is annihilated by the Lie subalgebra g ⊗ tC[[t]] for sufficiently large N ≥ 0, and on which 1 ∈ C ⊂ ĝκ acts as the identity. We will refer to objects of these category as modules at level κ. Let GrG = G((t))/G[[t]] be the affine Grassmannian of G. For each κ there is a category Dκ(GrG)–mod of κ-twisted right D-modules on GrG (see [BD]). We have the functor of global sections Γ : Dκ(GrG)–mod → ĝκ–mod, F 7→ Γ(GrG,F). Let κKil be the Killing form, κKil(x, y) = Tr(adg(x) ◦ adg(y)). The level κcrit = − 1 2κKil is called critical. A level κ is called positive (resp., negative, irrational) if κ = c · κKil and c+ 1 2 ∈ Q >0 (resp., c+ 12 ∈ Q , c / ∈ Q). It is known that the functor of global sections cannot be exact when κ is positive. In contrast, when κ is negative or irrational, the functor Γ is exact and faithful, as shown by A. Beilinson and V. Drinfeld in [BD], Theorem 7.15.8. This statement is a generalization for affine algebras of the famous theorem of A. Beilinson and J. Bernstein, see [BB], that the functor of global sections from the category of λ-twisted D-modules on the flag variety G/B is exact when λ− ρ is anti-dominant and it is faithful if λ− ρ is, moreover, regular. The purpose of this paper is to consider the functor of global sections in the case of the critical level κcrit. (In what follows we will slightly abuse the notation and replace the subscript κcrit simply by crit.) Unfortunately, it appears that the approach of [BD] does not extend to the critical level case, so we have to use other methods to analyze it. Our main result is that the functor of global sections remains exact at the critical level: