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.