Automorphisms of Hilbert schemes of points on a generic projective K3 surface

We study automorphisms of the Hilbert scheme of $n$ points on a generic projective K3 surface $S$, for any $n\ge 2$. We show that the automorphism group of $S^{[n]}$ is either trivial or generated by a non-symplectic involution and we determine numerical and divisorial conditions which allow us to distinguish between the two cases. As an application of these results we prove that, for any $n\ge 2$, there exist infinite values for the degree of $S$ such that $S^{[n]}$ admits a non-natural involution. This provides a generalization of results by Boissi`ere–Cattaneo–Nieper-Wisskirchen–Sarti for $n=2$.