# Which flag manifolds are symmetric? [closed]

A complex flag manifold is a quotient of a complex semi-simple Lie group by a parabolic subgroup (a subgroup which contains a Borel subgroup). Basic examples are complex projective space and the complex Grassmannians. Now all complex Grassmannian spaces are symmetric spaces, my question is: Which complex flag manifolds are symmetric?

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## closed as too localized by José Figueroa-O'Farrill, S. Carnahan♦Nov 16 '10 at 2:49

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en.wikipedia.org/wiki/Generalized_flag_variety#Symmetric_spaces I think that questions which can be answered by looking at the relevant wikipedia page are not appropriate to MO. – José Figueroa-O'Farrill Nov 15 '10 at 21:33
What is the relevant Wikipedia page? The entry on Symmetric spaces? – Abtan Massini Nov 15 '10 at 21:35
I agree with Jose's comment and would also recommend a comprehensive book such as Helgason's Differential Geometry, Lie Groups, and Symmetric Spaces for a thorough treatment of both real and complex Lie groups in this setting. (Like most Wikipedia articles, the reference list is unbalanced though the article itself has some useful information.) – Jim Humphreys Nov 15 '10 at 21:51
This page is probably closer: en.wikipedia.org/wiki/Hermitian_symmetric_space . Equivalent to the remark there: G/P is hermitian symmetric iff P is cominiscule. See also: mathoverflow.net/questions/20777/… . – Dave Anderson Nov 15 '10 at 21:58
@Dave: indeed. But there's a link to that page from the wiki page I quoted, and moreover, you get to learn also about which real flag manifolds are symmetric. – José Figueroa-O'Farrill Nov 15 '10 at 22:03