Rational Cherednik algebras and Hilbert schemes

Abstract

Let H c be the rational Cherednik algebra of type A n - 1 with spherical subalgebra U c = eH c e . Then U c is filtered by order of differential operators, with associated graded ring gr U c = C [ h ⊕ h * ] W where W is the n th 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 U c - mod; • the associated graded Z -algebra gr B has gr B - 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 U c 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 H c and its relationship to Hilb ( n ) .