Moduli of affine schemes with reductive group action

For a connected reductive group G and a finite-dimensional G-module V, we study the invariant Hilbert scheme that parameterizes closed G-stable subschemes of V affording a fixed, multiplicity-finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T-stable subscheme admitting a good quotient. The fibers of the quotient map classify affine G-schemes having a prescribed categorical quotient by a maximal unipotent subgroup of G. We show that V contains only finitely many multiplicity-free G-subvarieties, up to the action of the centralizer of G in GL(V). As a consequence, there are only finitely many isomorphism classes of affine G-varieties affording a prescribed multiplicity-free representation in their coordinate ring. Final version, to appear in Journal of Algebraic Geometry