  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.
  Third, we obtain a classification of simple $G$-modules $V$ with polynomial
algebras of $U'$-invariants (for $G$ simple).
