Hochschild cohomology of Hilbert schemes of points on surfaces

We compute the Hochschild cohomology of Hilbert schemes of points on surfaces
and observe that it is, in general, not determined solely by the Hochschild
cohomology of the surface, but by its "Hochschild-Serre cohomology": the
bigraded vector space obtained by taking Hochschild homologies with
coefficients in powers of the Serre functor. As applications, we obtain various
consequences on the deformation theory of the Hilbert schemes; in particular,
we recover and extend results of Fantechi, Boissi\`ere, and Hitchin.
Our method is to compute more generally for any smooth proper algebraic
variety $X$ the Hochschild-Serre cohomology of the symmetric quotient stack
$[X^n/\mathfrak{S}_n]$, in terms of the Hochschild-Serre cohomology of $X$.