Rational Cherednik algebras and Hilbert schemes

Let Hc be the rational Cherednik algebra of type An-1 with spherical subalgebra Uc = eHce. Then Uc is filtered by order of differential operators, with associated graded ring gr Uc = ℂ[h ⊕ h*]W where W is the nth symmetric group. We construct a filtered ℤ-algebra B such that, under mild conditions on c: • the category B-qgr of graded noetherian B-modules modulo torsion is equivalent to Uc-mod; • the associated graded ℤ-algebra gr B has grB-lqgr ≃ coh Hilb(n), the category of coherent sheaves on the Hilbert scheme of points in the plane. This can be regarded as saying that Uc simultaneously gives a non-commutative deformation of h ⊕ h*/W and of its resolution of singularities Hilb(n) → h ⊕ h*/W. As we show elsewhere, this result is a powerful tool for studying the representation theory of Hc and its relationship to Hilb(n). © 2005 Elsevier Inc. All rights reserved.