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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 19th 2014
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   Author: David_Corfield
   Format: MarkdownItexThat'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]] $G/T\cong G^{\mathbb{C}}/B$ where $G$ is a [[compact Lie group]], $T$ its [[maximal torus]], $G^{\mathbb{C}}$ the [[complexification]] of $G$, which is a complex semisimple group, and $B\subset G^{\mathbb{C}}$ 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 [[parabolic subgroup|parabolics]] which includes, for example, [[Grassmannians]]. The 'larger family' are the generalized flag varieties, no?

   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 $G/T\cong G^{\mathbb{C}}/B$ where $G$ is a compact Lie group, $T$ its maximal torus, $G^{\mathbb{C}}$ the complexification of $G$, which is a complex semisimple group, and $B\subset G^{\mathbb{C}}$ 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?

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeDec 19th 2014
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   Author: zskoda
   Format: MarkdownItexDepends 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.

   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.