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[script h sign ⊕ script h sign *] W, where W is the nth symmetric group. Using the ℤ-algebra construction from [GS], it is also possible to associate to a filtered H c-or Uc-module M a coherent sheaf φ̂(M) on the Hubert 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) → script h sign ⊕ script h sign */ W. For example, we prove the following. If c = 1/n so that Lc(triv) is the unique one-dimensional simple Hc-module, then φ̂(eLc(triv)) ≅ script O sign Zn, where Zn = Τ-1(0) is the punctual Hubert 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 that coincides with that on H0(Z n, ℒk), where ℒ ≅ script O sign Hilb(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 φ̂(eδc(μ)) equals Σλ Kμλ[Zλ], where Kμλ, are Kostka numbers and the Zλ are (known) irreducible components of Τ-1 (h/ W).