Recollement of deformed preprojective algebras and the Calogero–Moser correspondence

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 $A_1(\mathbb C)$; (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 naturally 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 $\mathbb A_\infty$-modules over $A_1$ to a more familiar setting of representation theory.