Intersection of conjugacy classes with Bruhat cells in Chevalley groups. (Q1880129)

Let \(G\) be a Chevalley group, i.e., \(G\) is the group of points \(\widetilde G(K)\), where \(\widetilde G\) is a simple and simply-connected algebraic group that is defined and quasi-split over a field \(K\). Therefore, one can define a Tits system \((G,B,N,S)\). In this paper, the authors investigate properties of intersections of Bruhat cells \(B\dot wB\) of \(G\) with conjugacy classes \(C\) of \(G\), in particular, they consider the question, when is \(B\dot wB\cap C\neq\emptyset\).