Cohomology of a Hamiltonian $T$-space with involution

Abstract

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.