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Answer by Allen Knutson for Orbits of Root Vectors
If $\alpha$ is a long root (e.g. if $\mathfrak g$ is simply-laced), then $e_\alpha$ is in the $G$-orbit of the high weight vectors. Projectively, its orbit looks like $G/P_\Theta$ where $P_\theta$ is...
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Let $G$ be a connected, simply-connected complex semisimple Lie group with Lie algebra $\frak{g}$. Fix a maximal torus $T\subseteq G$, and let...
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