Equivariant categories of symplectic surfaces and fixed loci of Bridgeland moduli spaces

Given an action of a finite group G on the derived category of a smooth projective variety X, we relate the fixed loci of the induced G-action on moduli spaces of stable objects in D b (Coh(X)) with moduli spaces of stable objects in the equivariant category D b (Coh(X)) G . As an application, we obtain a criterion for the equivariant category of a symplectic action on the derived category of a symplectic surface to be equivalent to the derived category of a surface. This generalizes the derived McKay correspondence and yields a general framework for describing fixed loci of symplectic group actions on moduli spaces of stable objects on symplectic surfaces.