Restriction of the Moment Map to Certain Non-lagrangian Submanifolds

Let T × M → M be a Hamiltonian torus action on a connected symplectic manifold M for which the associated moment map Φ : M → t∗ is proper as a map into a convex open set ρ ⊆ t∗. We consider a closed submanifold Q of M and show that under certain local conditions on Q one has Φ(Q) = Φ(M). We apply this result in the special case that Q arises as the fixed point set of some involution σ on M which is not necessarily antisymplectic.