Weight spaces of invariants of certain unipotent group actions (Q910857)

Let G be a complex semisimple algebraic group, T a maximal torus in G, R the root system, S a closed subset of the set \(R^+\) of positive roots, and \(H=U_ S\) the corresponding standard unipotent subgroup of G. Let \({\mathbb{C}}[G]\) be the algebra of regular functions on G and \({\mathbb{C}}[G]^ H\) the subalgebra of invariants in \({\mathbb{C}}[G]\) for the action of H on G by right translations. The equivalence of the following two conditions is proved: (i) dim \(V^ H_{\chi}\leq 1\) for all finite dimensional rational irreducible G-modules V and for all characters \(\chi\) of T; (ii) the set \(R^+\setminus S\) is linearly independent over \({\mathbb{Q}}\). The implication (ii)\(\Rightarrow (i)\) had been shown previously by \textit{F. Grosshans} by different methods [Math. Z. 193, 95-103 (1986; Zbl 0579.14015)]. It was shown by Grosshans that (i) implies finite generation of \({\mathbb{C}}[G]^ H\) (it was conjectured by the reviewer and Pommerening that \({\mathbb{C}}[G]^ H\) is always finitely generated but this conjecture is still open).