  Let $M$ be a compact, connected symplectic manifold with a Hamiltonian action
of a compact $n$-dimensional torus $G=T^n$. Suppose that $\sigma$ is an
anti-symplectic involution compatible with the $G$-action. The real locus of
$M$ is $X$, the fixed point set of $\sigma$. Duistermaat uses Morse theory to
give a description of the ordinary cohomology of $X$ in terms of the cohomology
of $M$. There is a residual $\G=(\Zt)^n$ action on $X$, and we can use
Duistermaat's result, as well as some general facts about equivariant
cohomology, to prove an equivariant analogue to Duistermaat's theorem. In some
cases, we can also extend theorems of Goresky-Kottwitz-MacPherson and
Goldin-Holm to the real locus.
