Invariant-theoretic properties of the derived group of the maximal unipotent subgroup

Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. In my talk, I am going to speak about actions of $U'$ on affine $G$-varieties. First, we consider the algebra of $U'$ invariants on $G/U$. We show 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 $V$, the subspace of fixed vectors $V^{U'}$ is a cyclic $U/U'$-module. Second, we study "symmetries" of Poincare series for $U'$- invariants on affine conical $G$-varieties. Here the results are very similar to those for the algebras of $U$-invariants. Third, I describe a classification of the simple $G$-modules $V$ with polynomial algebras of $U'$-invariants (for $G$ simple).