# Projective spaces and flag manifolds

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 $$\mathbb CP^2=S/H$$?

• The group is $SL_3$, and any maximal parabolic in it. – Sasha Nov 22 '18 at 18:52
• @Sasha So is it true in general that all projective spaces are flag manifolds? – Amrat A Nov 22 '18 at 19:03
• Yes, it is true in general. – Sasha Nov 22 '18 at 19:12

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 $$d_1 = 1$$.