On Generalized Moment Maps for Symplectic Compact Group Actions

A generalized moment map is proposed for arbitrary symplectic actions of compact connected Lie groups on closed symplectic manifolds, in the spirit of the circle -valued maps introduced by D. McDuff in the case of non-Hamiltonian circle actions. We study equivariance properties of generalized moments, show that they allow reduction procedures, and obtain in the torus case a version of the Atiyah-Guillemin-Sternberg convexity theorem. As illustration, we reformulate a proof of M.K. Kim that "complexity one" symplectic torus actions are Hamiltonian, and give a symplectic proof of the finiteness of certain symmetry groups of compact oriented surfaces.