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 $μ$. Suppose there exists a symplectic involution $θ:M\to M$, such that $μ\circθ=-μ$. Assuming that 0 is a regular value of $μ$, we calculate the trace of the action of $θ$ on the cohomology of $M$ in terms of the trace of the action of $θ$ on the symplectic reduction $μ^{-1}(0)/T$ of $M$. This result generalizes a theorem of R. Stanley, who considered the case when $M$ was a toric variety.