Abstract. In this paper, we introduce baby Verma modules for symplectic reflection algebras of complex reflection groups at parameter t = 0 (the so-called rational Cherednik algebras at parameter t = 0) and present their most basic properties. As an example, we use baby Verma modules to answer several problems posed by Etingof and Ginzburg, [5], and give an elementary proof of a theorem of Finkelberg and Ginzburg, [6].

Abstract. We introduce a new class of algebras called Poisson orders. This class includes the symplectic reflection algebras of Etingof and Ginzburg, many quantum groups at roots of unity, and enveloping algebras of restricted Lie algebras in positive characteristic. Quite generally, we study this class of algebras from the point of view of Poisson geometry, exhibiting connections between their representation theory and some well-known geometric constructions. As an application, we employ our results in the study of symplectic reflection algebras, completing work of Etingof and Ginzburg on when these algebras are finite over their centres, and providing a framework for the study of their representation theory in the latter case.

Let Hc be the rational Cherednik algebra of type A n−1 with spherical subalgebra Uc = eHce.Then Uc is filtered by order of differential operators, with associated graded ring grW 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 Uc-mod;• the associated graded Z-algebra gr B has gr B-qgr ≃ Coh Hilb(n), the category of coherent sheaves on the Hilbert scheme of points in the plane.1991 Mathematics Subject Classification. 14C05, 32S45,16S80, 16D90.

Abstract. For a Weyl group, W , and its reflection representation h, we find the character and Hilbert series for a quotient ring of C[h ⊕ h * ] by an ideal containing the W -invariant polynomials without constant term. This confirms conjectures of Haiman.

Abstract. Let Hc be the rational Cherednik algebra of type A n−1 with spherical subalgebra Uc = eHce.Then Uc is filtered by order of differential operators with associated graded ring gr Uc = 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 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 relate it to Hilb(n) and to the resolution of singularities τ : Hilb(n) → h ⊕ h * /W . For example, we prove: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 H 0 (Zn, L k ), where L ∼ =