The modality of a Borel subgroup in a simple algebraic group of type $E_8$

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $k$, where $\mathrm{char}k$ is either 0 or a good prime for $G$. We consider the modality $\mathrm{mod}(B:\mathfrak{u})$ of the action of a Borel subgroup $B$ of $G$ on the Lie algebra $\mathfrak{u}$ of the unipotent radical of $B$, and report on computer calculations used to show that $\mathrm{mod}(B:\mathfrak{u})=20$, when $G$ is of type ${\mathrm{E}}_8$. This completes the determination of the values for $\mathrm{mod}(B:\mathfrak{u})$ for $G$ of exceptional type.

3 pages