# Name for a class of parabolic subgroups

This is a reference request for a (the) name of the following class of parabolic subgroups of a complex simple Lie group $G$:

Recall that parabolic subgroups of $G$, containing fixed Borel subgroup, are determined by faces $c$ of the positive Weyl chamber. Interiors of some of these faces contain coroots of the group $G$. Is there a name for the class of parabolic subgroups $P$ and the corresponding partial flag manifolds $G/P$ coming from such faces?

(I do not know the name even in the special case when such $P$ is maximal parabolic. Such maximal parabolics exist except for the type $A$ and $D_3$ groups: Most of the time they correspond to the highest (co)root.) My tentative name for $P$'s and $G/P$'s is "root type parabolic subgroups and partial flag manifolds". In a paper I am cowriting, such subgroups and manifolds behave better than other parabolics.

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Are those the same as minuscule or cominuscule parabolic subgroups? –  Piotr Achinger Apr 2 '13 at 1:10
Piotr: No, they are neither. –  Misha Apr 2 '13 at 2:27
I would suggest to call this $G/P$ the adjoint Grassmannian. Just because $G/P$ for maximal $P$ is called a $G$-Grassmannian and when $P$ corresponds to the highest root one has $G/P \subset P({\mathfrak{g}})$. –  Sasha Apr 2 '13 at 3:02

A classification of ''partial flag manifolds'' in terms of Painted Dynkin diagrams is given for example in the article

D. Alekseevsky, A. Arvanitoyeorgos: Riemannian flag manifolds with homogeneous geodesics. Trans. Amer. Math. Soc. 359 (2007), no. 8, 3769–3789

as well as in

M. Bordemann, M. Forger and H. Römer: Homogeneous Kähler manifolds: paving the way towards new supersymmetric sigma models. Comm. Math. Phys. 102 (1986), no. 4, 605–617.

Actually there are 4 big infinite families $G^{\mathbb{C}}/P\cong G/K$ corresponding to the classical simple Lie groups $SU(n+1)$, $SO(2n+1)$, $Sp(n)$, $SO(n)$, and 101 non-isomorphic exceptional flag manifolds, corresponding to an exceptional simple Lie group $G_2$, $F_4$, $E_6$, $E_7$, $E_8$. We can encode all these flag manifolds by using the painted Dynkin diagrams (which actually are used to classify all parabolic subalgebras--see for example the book: Parabolic Geometries I, Mathematical Surveys and Monographs, Vol 154).

However it is difficult to give names on these homogeneous spaces, not only due to their large totality (they form a class of manifolds much larger than compact symmetric spaces), but also since partial flag manifolds having different geometric properties depending on their second Betti number, i.e. the number of the black roots which we paint black in the Dynking diagram of $G$. On some low-dimensional cases and mainly for compact hermitian symmetric spaces (which are partial flag manifolds with second Betti number 1), we can use some terminology coming from Grassmannians.

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Math3.14159: It is an interesting answer and maybe I should ask Alexeevsky if he has a name for what I am interested in. Note, however, that I am not asking for names of all partial flag manifolds, but only for the very special ones I described. –  Misha Apr 2 '13 at 2:30