Throughout this paper, G denotes a simple and simply connected algebraic group over C of rank r and H is a Cartan subgroup, with Lie algebras g = LieG and h = LieH. Let R be the root system of the pair (G,H), W the Weyl group, and Λ ⊆ h the coroot lattice. Fix once and for all a positive Weyl chamber, i.e. a set of simple roots ∆. The geometric invariant theory quotient of g by the adjoint acti...