Let $V$ be a finite-dimensional vector space over an algebraically closed field $F$, and $\dim V=d$. Let $G=GL(V)$, $P$ is the parabolic subgroup of $G$. Let $\mathcal{F}$ be the set of all partial flags in $V$.
Consider the diagonal action of $G$ on $\mathcal{F} \times \mathcal{F}$, and $\mathcal{O}$ is an arbitrary orbit of this action.
What is the Zariski closure of $\mathcal{O}$?
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$\begingroup$ For subgroup $H \subset G$, I think $G$ orbit on $G/H \times G/H$ are the same as $H$ orbits on $G/H$. Because closure of orbits are union of orbits, it should boils down to the usual answer for closure of $B$ orbits in $G/B$, i.e parametrised by the Weyl group, and the closure of $O_w$ is the union of $O_z$ with $z \geq w$. (for parabolic it should work similarly with $W$ replaced by $W_I$). $\endgroup$– Nicolas HemelsoetCommented Aug 26, 2022 at 9:33
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$\begingroup$ @NicolasHemelsoet Thank you for your comment!!! I wonder if there is any reference for me to get more details? $\endgroup$– zhichengzhangCommented Aug 26, 2022 at 10:37
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$\begingroup$ $P$ orbits in $G/Q$ for $P,Q$ parabolic is well understood (look for "generalised Bruhat decomposition"). It's probably somewhere in Borel's book on linear algebraic groups. $\endgroup$– Nicolas HemelsoetCommented Aug 26, 2022 at 13:30
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$\begingroup$ @NicolasHemelsoet Thank you! $\endgroup$– zhichengzhangCommented Aug 27, 2022 at 0:35
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