New derived autoequivalences of Hilbert schemes and generalised Kummer varieties

We show that for every smooth projective surface X and every n≥2 the push-forward along the diagonal embedding gives a Pn−1-functor into the Sn-equivariant derived category of X^n. Using the Bridgeland--King--Reid--Haiman equivalence this yields a new autoequivalence of the derived category of the Hilbert scheme of n points on X. In the case that the canonical bundle of X is trivial and n=2 this autoequivalence coincides with the known EZ-spherical twist induced by the boundary of the Hilbert scheme. We also generalise the 16 spherical objects on the Kummer surface given by the exceptional curves to n^4 orthogonal Pn−1-Objects on the generalised Kummer variety.