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 $U^{\mathrm{\prime }}$ the derived group of $U$. We study actions of $U^{\mathrm{\prime }}$ on affine $G$-varieties. First, we consider the algebra of $U^{\mathrm{\prime }}$ invariants on $G\mathrm{/}U$. We prove that $k[G\mathrm{/}U{]}^{U^{\mathrm{\prime }}}$ is a polynomial algebra of Krull dimension $2r$, where $r=rk(G)$. A related result is that, for any simple finite-dimensional $G$-module $V$, $V^{U^{\mathrm{\prime }}}$ is a cyclic $U\mathrm{/}{U}^{\mathrm{\prime }}$-module. Second, we study “symmetries” of Poincare series for $U^{\mathrm{\prime }}$-invariants on affine conical $G$-varieties. Third, we obtain a classification of simple $G$-modules $V$ with polynomial algebras of $U^{\mathrm{\prime }}$-invariants (for $G$ simple).