Abelian Localization for Cherednik Algebras

The goal of this talk is to prove an analog of the Beilinson-Bernstein localization theorem for Cherednik algebras. Strictly speaking we will only do this for categories O, in fact, the localization theorem for all modules follows from here. Let us recall the notation and some definitions. We consider the reflection representation h of the symmetric group Sn. By X we denote the “normalized” Hilbert scheme X, a resolution of X0 := (h⊕ h)/Sn, we write π for the Hilbert-Chow morphism X → X0. The structure sheaf OX has a two-parametric deformation Aun, a sheaf of algebras over C[z, ~], where z is a parameter of a commutative deformation X̃. We have a two dimensional torus Th × Tc (h for “Hamiltonian”, c for “contracting”) acting on X and on Aun. We can consider the specialization Aλ of Aun to ~ = 1, z = λ. It still carries a Th-action. We also consider the algebra S(h⊕h)#Sn and its two-parametric deformationHun over C[c, t] that again comes equipped with a Th × Tc-action. We consider the specialization H1,c of Hun to numerical parameters. Let P denote the Procesi bundle on X, a Th×Tc-equivariant vector bundle constructed in Gufang’s talk. Let P̃~ be its deformation to a right Aun-module (note that this is different from the previous lecture, where we used a deformation to a left module). We write H loc un for its endomorphism sheaf. This is a sheaf of C[z, ~]-algebras or of C[c, t]algebras, where c 7→ −z, t 7→ ~ (well, there is another choice of the map, and I’m not 100% sure what one needs to take...). As we have seen in the previous lecture, Γ(H loc un ) = Hun, a Th × Tc-equivariant isomorphism of C[c, t]-algebras. We remark that P̃~ ⊗Aun • is an equivalence of Coh(Aun) and Coh(H loc un ). We also have functors Γ : Coh(H loc un ) → Hun -mod of taking global sections and Loc : Hun -mod → Coh(H loc un ), the localization functor given by N 7→ H loc un ⊗Hun N . We also consider the specializations of these functors to numerical parameters. We have similar functors between Coh(H loc 1,c ), H1,c -mod. Let us denote those by Γ1,c,Loc1,c, they are obtained from Γ,Loc by specialization to numerical values of parameters. We also remark that RΓ and LLoc are mutually inverse derived equivalences. This was basically established in the previous lecture (with slightly different functors).