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comment Closed orbits of complete flags in $\mathbb{C}^n$
Yes, $O(n)$ here is really $O(\mathbb C^n, B)$, as stated in the OP.
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comment Closed orbits of complete flags in $\mathbb{C}^n$
@Mariano: No, it's not: if $g$ is in $O(2)$ then $g \cdot i = (\det g) i = \pm i$.
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revised Closed orbits of complete flags in $\mathbb{C}^n$
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answered Closed orbits of complete flags in $\mathbb{C}^n$
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