  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.
