Invariant functions on symplectic representations

Let G be a connected reductive group. In this paper we are studying the invariant theory of symplectic G-modules. Our main result is that the invariant moment map is equidimensional. We deduce that the categorical quotient is a fibration over an affine space with rational generic fibers. Of particular interest are those modules for which the generic orbit is coisotropic. We prove that they are cofree. This result has been used in another paper (math.SG/0505268) to classify all these modules. Our main tool is a symplectic version of the local structure theorem.