On the endomorphisms of Weyl modules over affine Kac-Moody algebras at the critical level

TL;DR

This paper provides a new proof that the endomorphism algebra of Weyl modules at the critical level is isomorphic to functions on monodromy-free opers, connecting representation theory with geometric structures.

Contribution

It offers an independent, concise proof of a key isomorphism relating Weyl module endomorphisms to geometric objects, expanding understanding of affine Kac-Moody algebra representations.

Findings

Endomorphism algebra is isomorphic to functions on monodromy-free opers.

The proof is derived from shift of argument subalgebras.

Clarifies the structure of Weyl modules at the critical level.

Abstract

We present an independent short proof of the main result of arXiv:0706.3725 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 arXiv:0712.1183 about the shift of argument subalgebras.