Actions of the Derived Group of a Maximal Unipotent Subgroup on G-Varieties

Let U be a maximal unipotent subgroup of a connected semisimple group G and U' the derived group of U. We study actions of U' on affine G-varieties. First, we consider the algebra of U' invariants on G/U. We prove that k[G/U](U') is a polynomial algebra of Krull dimension 2r, where r = rk(G). A related result is that, for any simple finite-dimensional G-module V, V-U' is a cyclic U/U'-module. Second, we study "symmetries" of Poincare series for U'-invariants on affine conical G-varieties. The results we obtain are very similar to those for the algebras of U-invariants. Third, we obtain a classification of simple G-modules V with polynomial algebras of U'-invariants (for G simple).