On a problem of Pommerening in invariant theory

Abstract

Hochschild and Mostow [3] have proved the finite generation of the algebra of invariants for the regular action of the unipotent radical of a parabolic subgroup on the polynomial functions on a complex semisimple algebraic group G. Pommerening [S] used combinatorial techniques to show that if G = SL, then we may remove any set of simple roots from the unipotent radical, and the algebra of invariants will still be finitely generated. In this paper we partially generalize Pommerening’s theorem to arbitrary semisimple groups G. In the proof we classify in terms of Dynkin diagrams, the cases for which the results of Grosshans [2] are applicable. Let R be the root system relative to a maximal torus T of G, and let D be a basis of R (which we identify with the Dynkin diagram of R). For a closed subset S of the set R + of positive roots, we denote by Us the corresponding standard unipotent subgroup of G, that is, U, = naES U,, where the product is taken in any fixed order and the U, are the root groups. For a subset I of D we denote S1 = R ‘\Z’I (so that U,, is the unipotent radical of a parabilic subgroup). Denote by C[G] the algebra of polynomial functions on G. The main result of this paper is the following.