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I asked this here https://math.stackexchange.com/questions/1441408/orbit-under-lie-group-and-projective-variety but did not get any answers, so I am asking here. I apologize if this is not appropriate for this site.

On https://en.wikipedia.org/wiki/Generalized_flag_variety#Highest_weight_orbits_and_homogeneous_projective_varieties there is a section which says

If G is a semisimple algebraic group (or Lie group) and V is a (finite dimensional) highest weight representation of G, then the highest weight space is a point in the projective space P(V) and its orbit under the action of G is a projective algebraic variety. This variety is a (generalized) flag variety, and furthermore, every (generalized) flag variety for G arises in this way.

I am interested in the case of Lie groups. Does anyone have a reference for this, which includes a proof? The bold parts are what really interest me.

I am also interested in whether the action of the real form $G_0 \subset G$ on the highest weight vector is also a projective algebraic variety.

Thanks.

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1 Answer 1

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This is part of the Borel-Weil theorem, more precisely it's Theorem 3 in Serre's 1954 exposition. As he notes there, the algebraicity you ask about is a consequence of Chow's theorem. Also, the orbit of the highest weight space under the compact real form $G_0$ is open in the $G$-orbit by dimension and closed by compactness, so it coincides with the $G$-orbit.

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  • $\begingroup$ Could you expand on the "open by dimension" bit? Thanks. $\endgroup$
    – nigel
    Commented Sep 19, 2015 at 21:41
  • $\begingroup$ @nigelvr See e.g. Theorem 5.8 on page 48 here. $\endgroup$ Commented Sep 19, 2015 at 22:19
  • $\begingroup$ @nigelvr Alternatively, if you already know that the orbit (flag variety) is simply connected, then transitivity of the compact real form on it also follows from Montgomery's Theorem. $\endgroup$ Commented Sep 19, 2015 at 22:25

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