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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.