Intersection of conjugacy classes with Bruhat cells in Chevalley groups: The cases $S{L}_n\left(K\right),G{L}_n\left(K\right)$

Abstract

Let $G$ be either $S{L}_n\left(K\right)$ or $G{L}_n\left(K\right)$ and let $C$ be a conjugacy class of $G$. We give conditions under which the intersection of a Bruhat cell of $G$ and a conjugacy class $C$ of $G$ is not empty and we show how to make some progress towards an algorithm which under certain conditions produces elements in this intersection. This is a continuation of an earlier investigation [E.W. Ellers, N. Gordeev, Intersection of conjugacy classes with Bruhat cells in Chevalley groups, Pacific J. Math. 214 (2) (2004) 245–261. MR 2004m:20091] and part of a larger program intended to deal with the above intersection problem for all Chevalley groups.