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 $\mu$. 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 character 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 and $\dim T = \frac{1}{2} \dim_{\mathbb{R}} M$.

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Published 24 June 2016