On intersections of conjugacy classes and Bruhat cells

TL;DR

This paper characterizes when conjugacy classes in a complex semi-simple Lie group intersect with Bruhat cells, linking the intersection to Bruhat order and an associated involution, with explicit results for special cases.

Contribution

It provides a criterion for non-empty intersections between conjugacy classes and Bruhat cells based on Bruhat order and involutions, including explicit descriptions for SL(n+1,C).

Findings

Intersection criterion based on Bruhat order and involution.

Explicit description of involution for SL(n+1,C).

Conditions for non-empty intersections when w is an involution.

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 $BwB^-$ 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.