On derived categories of K3 surfaces, symplectic automorphisms and the Conway group

In this note we interpret a recent result of Gaberdiel et al [7] in terms of derived equivalences of K3 surfaces. We prove that there is a natural bijection between subgroups of the Conway group Coo with invariant lattice of rank at least four and groups of symplectic derived equivalences of Db (X) of projective K3 surfaces fixing a stability condition. As an application we prove that every such subgroup G subset of Co-0 satisfying an additional condition can be realized as a group of symplectic automorphisms of an irreducible symplectic variety deformation equivalent to Hilb(n) (X) of some K3 surface.