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.

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

More generally, the

generalized flag varietyis 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?

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