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

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 ${\mathfrak{n}}^{\ast }$. By definition, ${\Omega }_{D,\xi }$ is the orbit of $f_{D,\xi }$, where $f_{D,\xi }$ is the sum of root covectors $e_{\alpha }^{\ast }$ 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 Andrè 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.

arXiv comment16 pages, 4 figures