ON TORIC HAMILTONIAN T-SPACES WITH ANTI-SYMPLECTIC INVOLUTIONS

The aim of this paper is to deal with the realization problem of a given Lagrangian submanifold of a symplectic manifold as the fixed point set of an anti-symplectic involution. To be more precise, let (X, omega, mu) be a toric Hamiltonian T-space, and let Delta = mu(X) denote the moment polytope. Let Tau be an anti-symplectic involution of X such that Tau maps the fibers of mu to (possibly different) fibers of mu, and let p0 be a point in the interior of Delta. If the toric fiber mu-1(p0) is real Lagrangian with respect to Tau, then we show that p0 should be the origin and, furthermore, Delta should be centrally symmetric.