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

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 nth symmetric group. Using the Z -algebra construction from [GS], it is also possible to associate to a filtered H c - or U c -module M a coherent sheaf Φ ∧ ( M ) on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of U c and H c , 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 L c ( triv ) is the unique one-dimensional simple H c -module, then Φ ∧ ( eL c ( triv ) ) ≅ O Z n , where Z n = 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 L c ( triv ) , gr eL c ( triv ) has a natural bigraded structure that coincides with that on H 0 ( Z n , L k ) , where L ≅ O 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 Φ ∧ ( e Δ c ( μ ) ) equals ∑ λ K μ λ [ Z λ ] , where K μ λ are Kostka numbers and the Z λ are (known) irreducible components of τ - 1 ( h / W )