Intersection of Conjugacy Classes with Bruhat Cells in Chevalley Groups

Let (G,B,N, S) be a Tits system. Some aspects of intersections of conjugacy classes ofG with Bruhat cellsBẇB have been investigated by several authors (see e.g., [St1], [K], [V] and [VS]). Here w ∈ W = N/(B ∩N) and ẇ ∈ N is a preimage of w with respect to the natural surjection N → W . In particular, it is desirable to learn how a conjugacy class C of G is related to those conjugacy classes Cw of W for which BẇB ∩ C = ∅, where w ∈ Cw. Here we deal with the case where G is a Chevalley group, i.e., G is the group of points G̃(K) of a simple algebraic group G̃ that is defined and quasi-split over a field K, thus G is a proper or a twisted Chevalley group (see [St2]). Therefore, one can define a Tits system (G,B,N, S), where S = {wαi | αi ∈ Π} for a simple root system Π corresponding to G ([St2] and [C1]). A crucial step to investigate intersections BẇB∩C was done by R. Steinberg [St1] who constructed the cross-section of regular conjugacy classes in BẇSB, where wS is a Coxeter element of W with respect to the fixed set of generators S of W, i.e., wS is a product of elements in S in any order, where each s ∈ S occurs exactly once. The next natural step is to consider intersections of regular classes with cells of the form BẇẇSẇ−1B. Here we prove the following: