The duality map on symplectic moduli spaces of sheaves

Duality defines an involution on the moduli space of slope-stable bundles with trivial determinant on a projective surface X. It extends to a birational involution on the moduli space of Gieseker semistable sheaves. When X is a K3 surface of Picard rank one, we characterize when the duality map is biregular and non-trivial in terms of the Mukai vector. Further analysis of the quotient by this involution yields new (singular) irreducible holomorphic symplectic varieties, with simply connected smooth locus and second Betti number 24. (Joint work in progress with R. Yamagishi.)