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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 19th 2014

    That’s not well put at flag variety, is it?

    More generally, the generalized flag variety is the complex projective variety obtained as the coset space G/TG /BG/T\cong G^{\mathbb{C}}/B where GG is a compact Lie group, TT its maximal torus, G G^{\mathbb{C}} the complexification of GG, which is a complex semisimple group, and BG B\subset G^{\mathbb{C}} is the Borel subgroup. It has a structure of a compact Kähler manifold. It is a special case of the larger family of coset spaces of semisimple groups modulo parabolics which includes, for example, Grassmannians.

    The ’larger family’ are the generalized flag varieties, no?

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeDec 19th 2014

    Depends on the author. You first have the usual falg variety GL(n)/B then you generalize to generalized G/B and finally to even more generalized G/P. Now the generalized ones are either the middle ones or the most general ones in the list, depending on the author.