Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves

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 grUc=C[h⊕h*]W, where W is the nth symmetric group. Using the Z-algebra construction from [GS], it is also possible to associate to a filtered Hc- or Uc-module M a coherent sheaf Φ∧(M) on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of Uc and Hc, and we relate it to Hilb(n) and to the resolution of singularities τ:Hilb(n)→h⊕h*/W. For example, we prove the following. • If c=1/n so that Lc(triv) is the unique one-dimensional simple Hc-module, then Φ∧(eLc(triv))≅OZn, where Zn=τ-1(0) is the punctual Hilbert scheme. • If c=1/n+k for k∈N, then under a canonical filtration on the finite-dimensional module Lc(triv), greLc(triv) has a natural bigraded structure that coincides with that on H0(Zn,Lk), where L≅OHilb(n)(1); this confirms conjectures of Berest, Etingof, and Ginzburg [BEG2, Conjectures 7.2, 7.3]. • Under mild restrictions on c, the characteristic cycle of Φ∧(eΔc(μ)) equals ∑λKμλ[Zλ], where Kμλ are Kostka numbers and the Zλ are (known) irreducible components of τ-1(h/W)