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Matthieu Romagny
Matthieu Romagny
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While the question and answers have explained different aspects of the story for complex groups and homogeneous spaces, it might be worthwhile to place the question in a broader context. The complex groups here can just as well be regarded as algebraic groups, with the underlying field algebraically closed of any characteristic. The picture is remarkably similar in all cases, though expressed in somewhat different language. In any case, for any homogeneous space the starting point is the ambient group acting naturally, but there may be further automorphisms.

Two classical references of interest are available online (though in French). Both are rather short but also somewhat sophisticated:

Demazure treated flag varieties in the context of algebraic geometry here. As he remarks, the complex situation had already been handled by Tits here.

The varieties or manifolds of interest all have the form $G/P$ but can be studied from different angles. Aside from this, I've added a tag tp emphasixeto emphasize the group-theoretic setting.

While the question and answers have explained different aspects of the story for complex groups and homogeneous spaces, it might be worthwhile to place the question in a broader context. The complex groups here can just as well be regarded as algebraic groups, with the underlying field algebraically closed of any characteristic. The picture is remarkably similar in all cases, though expressed in somewhat different language. In any case, for any homogeneous space the starting point is the ambient group acting naturally, but there may be further automorphisms.

Two classical references of interest are available online (though in French). Both are rather short but also somewhat sophisticated:

Demazure treated flag varieties in the context of algebraic geometry here. As he remarks, the complex situation had already been handled by Tits here.

The varieties or manifolds of interest all have the form $G/P$ but can be studied from different angles. Aside from this, I've added a tag tp emphasixe the group-theoretic setting.

While the question and answers have explained different aspects of the story for complex groups and homogeneous spaces, it might be worthwhile to place the question in a broader context. The complex groups here can just as well be regarded as algebraic groups, with the underlying field algebraically closed of any characteristic. The picture is remarkably similar in all cases, though expressed in somewhat different language. In any case, for any homogeneous space the starting point is the ambient group acting naturally, but there may be further automorphisms.

Two classical references of interest are available online (though in French). Both are rather short but also somewhat sophisticated:

Demazure treated flag varieties in the context of algebraic geometry here. As he remarks, the complex situation had already been handled by Tits here.

The varieties or manifolds of interest all have the form $G/P$ but can be studied from different angles. Aside from this, I've added a tag to emphasize the group-theoretic setting.

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Jim Humphreys
Jim Humphreys
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While the question and answers have explained different aspects of the story for complex groups and homogeneous spaces, it might be worthwhile to place the question in a broader context. The complex groups here can just as well be regarded as algebraic groups, with the underlying field algebraically closed of any characteristic. The picture is remarkably similar in all cases, though expressed in somewhat different language. In any case, for any homogeneous space the starting point is the ambient group acting naturally, but there may be further automorphisms.

Two classical references of interest are available online (though in French). Both are rather short but also somewhat sophisticated:

Demazure treated flag varieties in the context of algebraic geometry here. As he remarks, the complex situation had already been handled by Tits here.

The varieties or manifolds of interest all have the form $G/P$ but can be studied from different angles. Aside from this, I've added a tag tp emphasixe the group-theoretic setting.