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).