On the Endomorphisms of Weyl Modules over Affine Kac-moody Algebras at the Critical Level

We present an independent short proof of the main result of [FG07] that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. We derive this from the results of [FFR] about the shift of argument subalgebras. 1. Formulation of the main result 1.1. Weyl modules at the critical level. Let g be a simple Lie algebra, and ĝ be the corresponding affine Kac–Moody algebra. The Lie algebra ĝ is a central extension of the formal loop algebra g((t)) by one-dimensional center with generator 1 . The commutation relations are as follows: (1) [g1 ⊗ x(t), g2 ⊗ y(t)] = [g1, g2]⊗ x(t)y(t) + κc(g1, g2)Rest=0 x(t)dy(t) · 1, where κc is the invariant inner product on g defined by the formula (2) κc(g1, g2) = − 1 2 Trg ad(g1) ad(g2). Set ĝ+ = g[[t]] ⊂ ĝ and ĝ− = t −1g[t−1] ⊂ ĝ . Define the completion Ũ(ĝ) of U(ĝ) as the inverse limit of U(ĝ)/U(ĝ)(tg[[t]]), n > 0. The action of Ũ(ĝ) is well-defined on the category of discrete ĝ-modules, i.e., those in which every vector is annihilated by tg[[t]] for some n > 0. We set Ũκc(ĝ) = Ũ(ĝ)/(1 − 1). This algebra acts on discrete ĝ-modules of critical level (i.e., ĝ-modules on which the element K acts as unity). For a dominant integral weight λ of g , let πλ : g −→ EndC Vλ be the finitedimensional irreducible representation of g with the highest weight λ . One can naturally extend this representation to ĝ+ = g[[t]] by using the composition with the natural map g[[t]] −→ g corresponding to evaluation at t = 0. The Weyl module at the critical level with the highest weight λ is by definition the induced module Vλ := Ind bg bg+⊕C1 Vλ, Date: February 2008. 1 Supported by the grants RFBR 05-01-01007, RFBR 05-01-02934 and NSh-6358.2006.2. 2 Supported by DARPA and AFOSR through the grant FA9550-07-1-0543. 3 Supported by RFBR 05-01-02805-CNRSL-a, RFBR 07-01-92214-CNRSL-a, NSF grant DMS0635607 and Deligne 2004 Balzan prize in mathematics. L.R. gratefully acknowledges the Institute for Advanced Study for providing warm hospitality and excellent working conditions. 1 where 1 acts on Vλ as the identity. 1.2. Action of the center and monodromy-free opers. Consider the Langlands dual Lie algebra g whose Cartan matrix is the transpose of the Cartan matrix of g . Denote by G the group of inner automorphisms of g . In [FF, Fr05] the center Z(ĝ) of the completed enveloping algebra Ũκc(ĝ) at the critical level was identified with the algebra of polynomial functions on the space OpLG(D ×) of G-opers on the disc D = SpecC((t)). Let us recall the notion of opers which was introduced in [BD]. Fix a Cartan decomposition g = n ⊕ h ⊕ n−. The Cartan subalgebra h is canonically identified with h∗ . We denote by Π∨ the set of simple roots of g (which is the set of simple coroots of g). Set