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

TL;DR

This paper investigates the actions of the derived subgroup of a maximal unipotent subgroup on affine G-varieties, revealing polynomial invariants, module structures, and classifying modules with polynomial invariants.

Contribution

It provides new results on the structure of U'-invariants, their polynomial nature, and classifies simple G-modules with polynomial invariants, advancing understanding of unipotent subgroup actions.

Findings

U'-invariants form a polynomial algebra of dimension 2r

V^{U'}} is cyclic over U/U' for simple modules V

Classification of simple G-modules with polynomial U'-invariants

Abstract

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).