Recollement of Deformed Preprojective Algebras and the Calogero-Moser Correspondence

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/Γ$, where $\Gamma$ is a finite cyclic subgroup of $ \SL(2, \C) $.

16 pp., LaTex, to appear in Moscow Math. J.(2007)