Parabolic subgroups of Chevalley groups over a~semilocal ring

N. A. Vavilov

Zapiski Nauchnykh Seminarov POMI, Rings and linear groups, Tome 75 (1978), pp. 43-58

Résumé

Let $G$ be the Chevalley group over a commutative semilocal ring $R$ which is associated with a root system $\Phi$. The parabolic subgroups of $G$ are described in the work. A system $\sigma=(\sigma_\alpha)$ of ideals $\sigma_\alpha$ in $R$ ($\alpha$ runs through all roots of the system $\Phi$) is called a net of ideals in the commutative ring $R$ if $\sigma_\alpha\sigma_\beta\subset\sigma_{\alpha+\beta}$ for all those roots $\alpha$ and $\beta$ for which $\alpha+\beta$ is also a root. A net $\sigma$ is called parabolic if $\sigma_\alpha=R$ for $\alpha>0$. The main theorem: under minor additional assumptions all parabolic subgroups of $G$ are in bijective correspondence with all parabolic nets $\sigma$. The paper is related to two works of K. Suzuki in which the parabolic subgroups of $G$ are described under more stringent conditions. Bibl. 19 titles.