On intersections of conjugacy classes and bruhat cells Authors Kei Yuen Chan Department of Mathematics Hong Kong University Jiang-Hua Lu Department of Mathematics Hong Kong University Simon Kai-Ming To Department of Mathematics Hong Kong University Article

First Online: 16 April 2010 Received: 06 July 2009 Accepted: 09 January 2010 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 m _{C} ∈ 2 W associated to C . We prove that the element m _{C} 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 m _{C} explicitly for every conjugacy class C , and when w ∈ W ≌ S_{n+1} is an involution, we give an explicit answer to when C ∩ (BwB ) is nonempty.

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