  We show that for every smooth projective surface X and every $n\ge 2$ the
push-forward along the diagonal embedding gives a $P^{n-1}$-functor into the
$S_n$-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 $P^{n-1}$-Objects on the
generalised Kummer variety.
