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In this paper, we shall be concerned with obtaining positive results relating to Hilbert's fourteenth problem. In particular, our purpose, here, is to prove the following Theorem. Let k be an algebraically closed field. Let G be a reductive algebraic group over k and let U 1 be the unipotent radical of a parabolic subgroup of G. If X is any affine variety on which G acts (rationally), then k[X]Vl={fek[X]: u.f=f for all ueUl} is a finitely generated k-algebra.This theorem was proved for maximal unipotent subgroups in characteristic 0 in [2] and arbitrary characteristic in [1]. Some of the details of the latter proof will be used here so it is outlined in (2.1). The theorem, itself, was proved in characteristic 0 in [3] and in arbitrary characteristic for GL, in [8].A proof of the theorem in characteristic 0 is given in (2.2) since it points to a construction used later. This proof depends on the complete reducibility of reductive algebraic groups. For arbitrary characteristic, the argument requires the geometric properties of reductive groups found in [7; Theorem 3.5, p. 61], for example.

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Let k be an algebraically closed field of characteristic p 0. Let H be a subgroup of GLn(k). We are interested in the determination of the vector invariants of H. When the characteristic of k is 0, it is known that the invariants of d vectors, d n, are obtained from those of n vectors by polarization. This result is not true when char k = p > 0 even in the case where H is a torus. However, we show that the algebra of invariants is always the p-root closure of the algebra of polarized invariants. We also give conditions for the two algebras to be equal, relating equality to good filtrations and saturated subgroups. As applications, we discuss the cases where H is finite or a classical group.

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