G-endomorphisms of affine G-varieties which induce automorphisms of the invariant subrings of the coordinate rings

Abstract. Let G be a complex algebraic group and let X = Spec R be an ane G-variety. Let ' be a G-endomorphism of X which induces an automorphism of the invariant subring RG. We show that ' is a G-automorphism when G is unipotent and X is factorial. When G is reductive, we give some conditions under which ' is a G-automorphism. 1