Equivariant Moduli Theory on $K3$ Surfaces

In this paper we study equivariant moduli spaces of sheaves on a $K3$ surface $X$ under a symplectic action of a finite group. We prove that under some mild conditions, equivariant moduli spaces of sheaves on $X$ are irreducible symplectic manifolds deformation equivalent to Hilbert schemes of points on $X$ via a connection between Gieseker and Bridgeland moduli spaces, as well as the derived McKay correspondence.