Parabolic subgroups of Chevalley groups over a semilocal ring

Abstract

Let G be the Chevalley group over a commutative semilocal ring R which is associated with a root system Φ. The parabolic subgroups of G are described in the work. A system σ=(σα) of ideals σα in R (α runs through all roots of the system Φ) is called a net of ideals in the commutative ring R if σασβ⊂σα+β for all those roots α and β for which α+β is also a root. A net σ is called parabolic if σα=R for α>0. The main theorem: under minor additional assumptions all parabolic subgroups of G are in bijective correspondence with all parabolic nets σ. The paper is related to two works of K. Suzuki in which the parabolic subgroups of G are described under more stringent conditions.