Cohomology of a Hamiltonian T-space with involution

Let M be a compact symplectic manifold endowed with a Hamiltonian action of a compact torus T with a moment map μ. Suppose there exists a symplectic involution θ : M → M, such that μ ◦ θ = −μ. Assuming that 0 is a regular value of μ, we calculate the character 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 and dim T = ½ dim_ℝM.