Invariant functions on symplectic representations

We study invariants on symplectic representations of a connected reductive group. 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 representations where these fibers are points. We show that they are cofree. Our main tool is a symplectic version of the local structure theorem.