Rational covariants of reductive groups and homaloidal polynomials

Let G be a complex reductive group, V a G-module and f a nonconstant homogenous invariant polynomial on V. We investigate relations between the following properties:- The differential df: V -< V* is dominant;- The invariant f is homaloidal, i.e., df induces a birational map P(V) -< P(V*);- V is a stable representation, i.e., the generic G-orbit in V is closed.If f generates the invariants, we show that the properties are equivalent, generalizing results of Sato-Kimura on prehomogeneous vector spaces.