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



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> 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.
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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.



Flag varieties and orbit of highest weight vector
This is part of the Borel-Weil theorem, more precisely it's Theorem 3 in Serre's 1954 [exposition](http://www.numdam.org/item?id=SB_1951-1954__2__447_0). As he notes there, the algebraicity you ask about is a consequence of [Chow's theorem](http://www.ams.org/mathscinet-getitem?mr=33093). 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.


