Intersection of conjugacy classes with Bruhat cells in Chevalley groups: The cases SLn(K),GLn(K)

Let G be either SLn(K) or GLn(K) 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.