Actions of the derived group of a maximal unipotent subgroup on $G$-varieties

Abstract

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

arXiv comment23 pages