Verma Modules of Critical Level and Differential Forms on Opers

Let g be a simple finite-dimensional Lie algebra and ĝκ, where κ is an invariant inner product on g, the corresponding affine Kac-Moody algebra. Consider the vacuum module Vκ over ĝκ (see Section 2 for the precise definitions). According to the results of [FF, Fr], the algebra of endomorphisms of Vκ is trivial, i.e., isomorphic to C, unless κ = κc, the critical value. The algebra Endĝκc Vκc is canonically isomorphic to the algebra of functions on the space OpLg(D) of g–opers on the disc, where g is the Lie algebra that is Langlands dual to g (its Cartan matrix is the transpose of that of g). In this paper we consider the algebra of endomorphisms of Vκc in the derived category of (ĝκc , G[[t]])–modules (here G is the connected simply-connected algebraic group corresponding to g). This algebra can be realized as the relative cohomology