  Let $M$ be a compact symplectic manifold on which a compact torus $T$ acts
Hamiltonialy with a moment map $\mu$. Suppose there exists a symplectic
involution $\theta:M\to M$, such that $\mu\circ\theta=-\mu$. Assuming that 0 is
a regular value of $\mu$, we calculate the trace of the action of $\theta$ on
the cohomology of $M$ in terms of the trace of the action of $\theta$ on the
symplectic reduction $\mu^{-1}(0)/T$ of $M$. This result generalizes a theorem
of R. Stanley, who considered the case when $M$ was a toric variety.
