  Let $S$ be a smooth, quasi-projective complex surface with complex symplectic
form $\omega \in H^0(S, K_S)$. This determines a symplectic form $\omega_n$ on
the Hilbert scheme of points $S^{[n]}$ for $n \geq 1$. Let $\tau$ be an
anti-symplectic involution of $(S,\omega)$: an order two automorphism of $S$
such that $ \tau^*\omega=-\omega$. Then $\tau$ induces an anti-symplectic
involution on $(S^{[n]},\omega_n)$ and the fixed point set $(S^{[n]})^\tau$ is
a smooth Lagrangian subvariety of $S^{[n]}$. In this paper, we calculate the
mixed Hodge structure of $H^*( (S^{[n]})^\tau; \mathbb{Q})$ in terms of the
mixed Hodge structures of $H^*( S^\tau;\mathbb{Q})$ and of $H^*( S / \tau;
\mathbb{Q})$. We also classify the connected components of $(S^{[n]})^\tau$ and
determine their mixed Hodge structures. Our results apply more generally
whenever $S$ is a smooth quasi-projective surface, and $\tau$ is an involution
of $S$ for which $S^\tau$ is a curve.
