Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution

The Kostant convexity theorem for real flag manifolds is generalized to a Hamiltonian framework. More precisely, it is proved that if f f is the momentum mapping for a Hamiltonian torus action on a symplectic manifold M M and Q Q is the fixed point set of an antisymplectic involution of M M leaving f f invariant, then f ( Q ) = f ( M ) = f(Q) = f(M) = a convex polytope. Also it is proved that the coordinate functions of f f are tight, using "half-turn" involutions of Q Q .