Rook placements in $G_2$ and $F_4$ and associated coadjoint orbits
Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Lie algebra with root system $Φ$. A subset $D$ of the set $Φ^+$ 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 $ξ$ from $D$ to the set $\mathbb{C}^{\times}$ of nonzero complex numbers one can naturally assign the coadjoint orbit $Ω_{D,ξ}$ in the dual space $\mathfrak{n}^*$. By definition, $Ω_{D,ξ}$ is the orbit of $f_{D,ξ}$, where $f_{D,ξ}$ is the sum of root covectors $e_α^*$ multiplied by $ξ(α)$, $α\in D$. (In fact, almost all coadjoint orbits studied at the moment have such a form for certain $D$ and $ξ$.) It follows from the results of Andrè that if $ξ_1$ and $ξ_2$ are distinct maps from $D$ to $\mathbb{C}^{\times}$ then $Ω_{D,ξ_1}$ and $Ω_{D,ξ_2}$ do not coincide for classical root systems $Φ$. We prove that this is true if $Φ$ is of type $G_2$, or if $Φ$ is of type $F_4$ and $D$ is orthogonal.