A connectedness property of algebraic moment maps

Let K be a connected compact Lie group and M a Hamiltonian K-manifold, i.e., a symplectic K-manifold equipped with a moment map μ : M → k := (LieK). A theorem of Kirwan (implicitly in [Ki]) asserts: if M is connected and compact then the level sets of μ are connected. The purpose of this note is to prove such a statement in the category of algebraic varieties. First, we reformulate Kirwan’s theorem: consider the map ψ : M → k/K which is the composition of μ with the quotient map. For a point x ∈ k let H = Kx be its isotropy group and y = Kx ∈ k/K its orbit. Then the fiber ψ(y) is isomorphic to the fiber product K × μ(x). Thus, since both K/H and H are connected, the connectedness of the fibers of μ is equivalent to the connectedness of the fibers of ψ. This formulation is more suitable for the algebraic category. Let G be a connected reductive group (everything over C) and Z a Hamiltonian Gvariety with moment map μ : Z → g = (LieG). Let g//G := SpecC[g] be the categorical quotient and let ψ̃ : Z → g//G be the composition of μ with the quotient map. Since the latter is not an orbit map the connection between fibers of μ and ψ̃ is much more loose than in the differential category and we concentrate on ψ̃ from now on. The morphism ψ̃ is still not the right map, since sometimes not even its generic fibers are connected. An example is the action of G = SL2(C) on Z = C 2 ×(C) (this is the cotangent bundle of C). Then g//G = A and ψ̃(u, α) = α(u). In particular, the generic fiber of ψ̃ has two connected components. This is remedied by looking at the map ψ(u, α) := α(u) instead. Then ψ̃ is the composition of ψ with the finite map A → A : z 7→ z, the latter being responsible for the disconnected fibers. This construction can be generalized as follows: the morphism ψ̃ induces an homomorphism of algebras