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Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the Zariski-closed $G$-orbits on the variety of complete flags in $\mathbb{C}^n$ for these two cases?

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