Speaker: Gwyn Bellamy (U Glasgow)

Abstract: Via localization theorems à la Beilinson-Bernstein, representations of quantizations of symplectic singularities are equivalent to modules over sheaves of deformation-quantization algebras (DQ-modules) on symplectic resolutions of the singularity. This applies for instance to (spherical) rational Cherednik algebras and finite W-algebras as well as the primitive central quotients of enveloping algebras appearing in the original Beilinson-Bernstein theorem. Usually, the sympletic resolution is equipped with a Hamiltonian C*-action, who attracting locus is a Lagrangian (with modules supported on this Lagrangian belonging to geometric category O). I'll explain that it's possible to construct a "local generator" in geometric category O such that the bounded derived category of coherent DQ-modules is equivalent to the derived category of coherent modules over the dg-endomorphism ring of the generator. This is a generalization of the classical D-Omega duality of Kapranov, Beilinson-Drinfeld and Positselski and thus an example of filtered Koszul duality. This talk is based on recent joint work with Chris Dodd, Kevin McGerty and Tom Nevins.