On symplectic birational self-maps of projective hyperkähler manifolds of K3${}^{[n]}$-type

We prove that projective hyperkähler manifolds of K3${}^{[n]}$-type admitting a non-trivial symplectic birational self-map of finite order are isomorphic to moduli spaces of stable (twisted) coherent sheaves on K3 surfaces. Motivated by this result, we analyze the reflections on the movable cone of moduli spaces of sheaves and determine when they come from a birational involution.