Rook placements in $G_2$ and $F_4$ and associated coadjoint orbits

TL;DR

This paper classifies certain coadjoint orbits associated with rook placements in specific Lie algebra root systems, extending known results to the exceptional types G2 and orthogonal F4.

Contribution

It proves that distinct maps from rook placements produce different coadjoint orbits in G2 and orthogonal F4 root systems, generalizing previous classical results.

Findings

Distinct maps yield different coadjoint orbits in G2.

Distinct maps yield different coadjoint orbits in orthogonal F4.

Extends classical orbit classification to exceptional Lie types.

Abstract

Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Lie algebra with root system $\Phi$. A subset $D$ of the set $\Phi^+$ of positive roots is called a rook placement if it consists of roots with pairwise non-positive scalar products. To each rook placement $D$ and each map $\xi$ from $D$ to the set $\mathbb{C}^{\times}$ of nonzero complex numbers one can naturally assign the coadjoint orbit $\Omega_{D,\xi}$ in the dual space $\mathfrak{n}^*$. By definition, $\Omega_{D,\xi}$ is the orbit of $f_{D,\xi}$, where $f_{D,\xi}$ is the sum of root covectors $e_{\alpha}^*$ multiplied by $\xi(\alpha)$, $\alpha\in D$. (In fact, almost all coadjoint orbits studied at the moment have such a form for certain $D$ and $\xi$.) It follows from the results of Andr\`e that if $\xi_1$ and $\xi_2$ are distinct maps from $D$ to $\mathbb{C}^{\times}$ then $\Omega_{D,\xi_1}$ and $\Omega_{D,\xi_2}$ do not coincide for classical root systems $\Phi$. We prove that this is true if $\Phi$ is of type $G_2$, or if $\Phi$ is of type $F_4$ and $D$ is orthogonal.