# The quotient of a Lie group by the Levi factor of a parabolic subgroup

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.

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It would help to make the question a little more precise,. starting with the wording: "the group generated by the Levi factor of a parabolic subalgebra" should be something like "a Levi factor of a parabolic subgroup" (note that Levi factors are only determined up to conjugacy). – Jim Humphreys Feb 4 '11 at 14:59
P.S. These types of quotient spaces have been written about from many viewpoints in books and research papers over the years, for instance the flag variety of a compact Lie group. Classical treatments are found in books by Helgason and Wolf, but beyond that many books treat compact and other Lie groups in the spirit of differential (or algebraic) geometry. Spaces like the $n$-sphere have been studied as homogeneous spaces of special orthogonal groups, etc. The question as written is too broad. What books on Lie groups have you already looked at? – Jim Humphreys Feb 4 '11 at 23:08
The totality of the question is essentially "What is the status of the complete classification of all of the spaces of type G/L for simple G?" Perhaps you are telling me that a detailed answer might be too broad but I think that the question itself is not. – Q.Q.J. Feb 5 '11 at 1:00
If you want to classify them, how is that different for you from classifying parabolics, which are classified by subsets of the Dynkin diagram? Also, I'm confused by "Levi factor of parabolic ... compact case". For compact case, do you mean compact Lie groups? In which case I would think Levi factor = parabolic. – Allen Knutson Feb 5 '11 at 3:46
The Levi factor is the semisimple part of the parabolic... The imagined classification would ideally include a tabulation of the fundamental groups of all possible G/L for example. – Q.Q.J. Feb 5 '11 at 5:02

Exhausting account on homogeneous spaces of the form $G/P$ with $G$ semisimple and $P$ parabolic is given in this book. The relationship between $P$ and $L$ is also explained there in detail so presumably one may take these results as a starting point.