A Geometric Proof of the Feigin-Frenkel Theorem

TL;DR

This paper provides a geometric proof of the Feigin-Frenkel theorem, connecting the center of the critical level enveloping algebra to opers via Langlands duality and the geometric Satake correspondence.

Contribution

It offers a new geometric proof of the Feigin-Frenkel theorem, linking algebraic and geometric aspects of the Langlands program.

Findings

Establishes a geometric proof of the Feigin-Frenkel theorem.

Connects the Feigin-Frenkel isomorphism to Langlands duality.

Utilizes the geometric Satake theorem in the proof.

Abstract

We reprove the theorem of Feigin and Frenkel relating the center of the critical level enveloping algebra of the Kac-Moody algebra for a semisimple Lie algebra to opers (which are certain de Rham local systems with extra structure) for the Langlands dual group. Our proof incorporates a construction of Beilinson and Drinfeld relating the Feigin-Frenkel isomorphism to (more classical) Langlands duality through the geometric Satake theorem.