Let 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 C × of nonzero complex numbers one can naturally assign the coadjoint orbit Ω D,ξ in the dual space 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 ∈ 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 C × then Ω D,ξ1 and Ω D,ξ2 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.
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.
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