Suzuki functor at the critical level

TL;DR

This paper introduces a critical-level generalization of the Suzuki functor linking affine Lie algebras and rational Cherednik algebras, revealing new algebraic and geometric structures and their interrelations.

Contribution

It defines a new functor at the critical level, establishes a surjective homomorphism between centers, and explores geometric embeddings related to Calogero-Moser space.

Findings

Surjective homomorphism from the center of the affine algebra to the Cherednik algebra

Explicit computation of the functor on key modules

Embedding of Calogero-Moser space into the space of opers

Abstract

In this paper we define and study a critical-level generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t=0. We use this homomorphism to obtain several results about the functor. We compute it on Verma modules, Weyl modules, and their restricted versions. We describe the maps between endomorphism rings induced by the functor and deduce that every simple module over the rational Cherednik algebra lies in its image. Our homomorphism between the two centres gives rise to a closed embedding of the Calogero-Moser space into the space of opers on the punctured disc. We give a partial geometric description of this embedding.