Recollement of Deformed Preprojective Algebras and the Calogero-Moser Correspondence

TL;DR

This paper establishes functorial connections between modules over Weyl algebras, deformed preprojective algebras, and rational Cherednik algebras, revealing their common geometric parametrization via Calogero-Moser varieties.

Contribution

It constructs explicit functors linking module categories of these algebras, extending previous results and providing a unified geometric framework.

Findings

Identifies geometric parametrization of modules by Calogero-Moser varieties.

Constructs functors between different module categories.

Extends the framework to Kleinian singularities.

Abstract

The aim of this paper is to clarify the relation between the following objects: $ (a) $ rank 1 projective modules (ideals) over the first Weyl algebra $ A_1(\C)$; $ (b) $ simple modules over deformed preprojective algebras $ \Pi_{\lambda}(Q) $ introduced by Crawley-Boevey and Holland; and $ (c) $ simple modules over the rational Cherednik algebras $ H_{0,c}(S_n) $ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on $\A$-modules over $ A_1 $ to a more familiar setting of representation theory. In the last section we extend our construction to the case of Kleinian singularities $ \C^2/\Gamma $, where $ \Gamma $ is a finite cyclic subgroup of $ \SL(2, \C) $.