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

Abstract

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 ∼ =

Differential operators and Cherednik algebras

“…This also clarifies the relationship between the two approaches, for example showing that the characteristic cycles defined independently in [GS2] and [GG] are actually equal, thereby confirming a conjecture of [GG].…”

“…The key to [GS1,GS2] is the construction of a Z-algebra B = i≥j≥0 ( c+i P c+j ) endowed with a natural matrix multiplication. The ring B has a natural filtration induced from the differential operator filtration Γ on D(h reg ) * W , and the main result [GS1,Theorem 1.4] showed that if c + i is good for all i ∈ N, then the associated graded ring gr Γ B of B is the Z-algebra associated to Hilb n C 2 .…”

Quiver Varieties, Category for Rational Cherednik Algebras, and Hecke Algebras

“…This is true when ℓ = 1 by [23,Theorem 6.7] and it has been established for n = 1 and any ℓ in [30]. In 10.3…”

Rational Cherednik Algebras

“…By an important result of Heckman-Opdam, see [9], eH 1,c (S n )e − is a (U 1,c (S n ), U 1,c+1 (S n ))-bimodule and one can show it induces an equivalence The advantage of this construction is that one can apply Haiman's work directly. This leads in [46] to the calculation of the characteristic cycle of any object from O c (S n ), i.e. the support cycles in Hilb n (C 2 ) of the degeneration of the corresponding objects in X c ; one can also show that the image in X c of the U 1,c (S n )-module eH 1,c (S n ) is a deformation of the Procesi bundle P on Hilb n (C 2 ).…”