On intersections of conjugacy classes and Bruhat cells

Abstract

For a connected complex semi-simple Lie group $G$ 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 $Bw{B}^{-}$ is non-empty, 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 $\mc \in W$ associated to $C$. We study properties of the elements $\mc$. For $G = SL(n+1, \Cset)$, we describe $\mc$ explicitly for every conjugacy class $C$, and for the case when $w\in W$ is an involution, we also give an explicit answer to when $C\cap (BwB)$ is non-empty.

20 pages. Revised version. to appear in Transformation Groups

Cited by in corpus (1)

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