On intersections of conjugacy classes and bruhat cells

Abstract

For a connected semisimple algebraic group G over an algebraically closed field k and a fixed pair (B, B-) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB- is nonempty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution mC ∈ W associated to C. We prove that the element mC is the unique maximal length element in its conjugacy class in W, and we classify all such elements in W. For G = SL(n + 1; k), we describe mC explicitly for every conjugacy class C, and when w ∈ W ≊ Sn+1 is an involution, we give an explicit answer to when C ∩ (BwB) is nonempty. © 2010 Springer Science+Business Media, LLC.