Cohomology of quotients in real symplectic geometry

Given a Hamiltonian system $ (M,\omega, G,\mu) where $(M,\omega)$ is symplectic manifold, $G$ compact connected Lie group acting on with moment map \mu:M \rightarrow\mathfrak{g}^{*}$, then one may construct the quotient $(M//G, \omega_{red})$ $M//G := \mu^{-1}(0)/G$. Kirwan used norm-square of map, $|\mu|^2$, as G-equivariant Morse function $M$ to derive formulas for rational Betti numbers $M//G$. A real $(M,\omega, G,\mu, \sigma, \phi) along pair involutions $(\sigma:M \rightarrow M, \phi:G G) satisfying certain compatibility conditions. These imply that fixed point set $M^{\sigma}$ Lagrangian submanifold and $M^{\sigma}//G^{\phi} (\mu^{-1}(0) \cap M^{\sigma})/G^{\phi}$ \omega_{red})$. In this paper we prove analogues Kirwan's Theorems can be calculate $\mathbb{Z}_2$-Betti $. particular, (under appropriate hypotheses) $|\mu|^2$ restricts $G^{\phi}$-equivariantly perfect Morse-Kirwan over $\mathbb{Z}_2$ coefficients, describe its critical using explicit subsystems, equivariant formality $G^{\phi}$ $M^{\sigma}$, combine these results produce $M^{\sigma}//G^{\phi}$.