A new look at the decomposition of unipotents and the normal structure of Chevalley groups

Abstract:

The paper continues a series of publications on the decomposition of unipotents in a Chevalley group $\mathrm {G} (\Phi ,R)$ over a commutative ring $R$ with a reduced irreducible root system $\Phi$. Fix $h\in \mathrm {G} (\Phi ,R)$. An element $a\in \mathrm {G} (\Phi ,R)$ is said to be “good” if it belongs to the unipotent radical of a parabolic subgroup and the conjugate to $a$ by $h$ lies in another parabolic subgroup (all parabolic subgroups are assumed to contain the same split maximal torus). The “decomposition of unipotents” method is a representation of an elementary root unipotent element as a product of “good” elements. Decomposition of unipotents implies a simple proof of normality for the elementary subgroup and the standardness for the normal structure of $\mathrm {G} (\Phi ,R)$. However, such a decomposition is available not for all root systems. In the paper, it is shown that to prove the standardness of the normal structure it suffices to find one “good” element for the generic element of the group scheme $\mathrm {G}(\Phi , \cdot)$. Also, some “good” elements are constructed. The question as to whether and when good elements span the elementary subgroup will be considered in a subsequent article of the series.