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

Abstract

The current article continues a series of papers on decomposition of unipotents and its applications. Let $G(\Phi,R)$ be a Chevalley group with a reduced irreducible root system $\Phi$ over a commutative ring $R$. Fix $h\in G(\Phi,R)$. Call an element $a\in G(\Phi,R)$ good, if it lies in the unipotent radical of a parabolic subgroup whereas the conjugate to $a$ by $h$ belongs to another proper parabolic subgroup (here we assume that all parabolics contain a given split maximal torus). Decomposition of unipotents is a representation of a root unipotent element as a product of good elements. Existence of such a decomposition implies a simple proof of the normality of the elementary subgroup and description of the normal structure of $G(\Phi,R)$. However, the decomposition is available not for all root systems. In the current article we show that for the proof of the standard normal structure it suffices to construct only one good element for the generic element of the scheme $G(\Phi,_-)$, and construct such a good element. The question whether good elements generate the whole elementary group will be addressed in a separate paper (the first version of the article was in Russian).