I am interested in some references on the quotient spaces obtained by quotienting G, a simple Lie group, by L, the group generated by the Levi factor of a parabolic subalgebra.


Presumably the case where L is the maximal torus is understood?


I am mostly interested in the compact case.



The quotient of a Lie group by the Levi factor of a parabolic subgroup
Exhausting account on homogeneous spaces of the form $G/P$ with $G$ semisimple and $P$ parabolic is given in this [book](http://www.ams.org/bookstore-getitem/item=surv-154). The relationship between $P$ and $L$ is also explained there in detail so presumably one may take these results as a starting point.


