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Is the complex projective space $\mathbb CP^2$ a flag variety? If yes, what are the complex semisimple Lie group $S$ and a parabolic subgroup $H$ such that $\mathbb CP^2=S/H$?

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    $\begingroup$ The group is $SL_3$, and any maximal parabolic in it. $\endgroup$ – Sasha Nov 22 '18 at 18:52
  • $\begingroup$ @Sasha So is it true in general that all projective spaces are flag manifolds? $\endgroup$ – Amrat A Nov 22 '18 at 19:03
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    $\begingroup$ Yes, it is true in general. $\endgroup$ – Sasha Nov 22 '18 at 19:12
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More generally, for any sequence $d_1 < d_2 < \dots < d_k < n$, the space of all partial flags $$V_1 \subset V_2 \subset \dots \subset V_k \subset \Bbb C^n$$

with $\dim V_i = d_i$ has a transitive $SL_n(\Bbb C)$-action. You can easily write down the stabiliser and more or less by definition it's a parabolic subgroup. Your question is the special case when $n=3, k=1$ and $d_1 = 1$.

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