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 n-th symmetric group. Using the Z-algebra construction from [GS] it is also possible to associate to a filtered Hcor Uc-module M a coherent sheaf Φ̂(M) on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of Uc and Hc, and relate it to Hilb(n) and to the resolution of singularities τ : Hilb(n) → h⊕ h∗/W . For example, we prove: • If c = 1/n, so that Lc(triv) is the unique one-dimensional simple Hc-module, then Φ̂(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), gr eLc(triv) has a natural bigraded structure which coincides with that on H0(Zn,Lk), where L ∼= OHilb(n)(1); this confirms conjectures of Berest, Etingof and Ginzburg. • Under mild restrictions on c, the characteristic cycle of Φ̂(e∆c(μ)) equals ∑ λ Kμλ[Zλ], where Kμλ are Kostka numbers and the Zλ are (known) irreducible components of τ −1(h/W ).