Invariants of a maximal unipotent subgroup and equidimensionality

Let U be a maximal unipotent subgroup of a semisimple group G. If G acts on an affine variety X, then it was proved by Hadžiev (1967) that there is a finitely generated k-algebra A such that kU[X]≃G(k[X]⊗A). It follows that kU[X] is finitely generated. This note contains two contributions to the theory of U-invariants. First, we obtain a relationship between the fibres of the quotient morphisms πU:X→X//U and πG:X×Spec(A)→(X×Spec(A))//G that contain T-fixed points. (Here T⊂NG(U) is a maximal torus of G.) For X conical, this implies that πU is equidimensional if and only if πG is. Second, we give a criterion of equidimensionality of πU for a class of varieties with a dense G-orbit (the so-called S-varieties of Vinberg and Popov).