Transformation Groups

, Volume 15, Issue 2, pp 243–260

On intersections of conjugacy classes and bruhat cells

Article

DOI: 10.1007/s00031-010-9084-7

Cite this article as:
Chan, K.Y., Lu, J. & Kai-Ming To, S. Transformation Groups (2010) 15: 243. doi:10.1007/s00031-010-9084-7

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 ∈ 2 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 wW ≌ Sn+1 is an involution, we give an explicit answer to when C ∩ (BwB) is nonempty.

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Kei Yuen Chan
    • 1
  • Jiang-Hua Lu
    • 1
  • Simon Kai-Ming To
    • 1
  1. 1.Department of Mathematics Hong Kong UniversityPokfulam RoadHong Kong