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Real flag manifolds (also known as R-spaces) can be defined in two ways which I believe are equivalent although some fine print may have escaped me:

  • as a quotient of a semisimple real Lie group $G$ by a parabolic subgroup (subgroup containing a Borel),

  • as an orbit of the isotropy representation (action of the point stabilizer on the tangent space) of the Riemann symmetric space associated to $G$.

Where can I find a explicit statement and proof of this equivalence, ideally in book form? Preferably with additional details on the connection between flag manifolds and Riemann symmetric spaces (e.g., on when one is the other, and conversely)?

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A real flag manifold is $M=G/P$ where $G$ is a real semisimple Lie group and $P$ is a parabolic subgroup of $G$. Here parabolic means that the Lie algebra $\mathfrak p$ contains a minimal parabolic subalgebra $\mathfrak p_{min}$ of $\mathfrak g$.

In order to define this, let $\mathfrak g=\mathfrak k+\mathfrak s$ be a Cartan decomposition, $\mathfrak a$ is a maximal Abelian subspace of $\mathfrak s$, $\mathfrak g =\mathfrak g_0+\sum_{\lambda\in\Sigma} \mathfrak g_\lambda$ be the root decomposition with respect to $\mathfrak a$. We have $\mathfrak g_0=\mathfrak m+\mathfrak a$ where $\mathfrak m$ is the centralizer of $\mathfrak a$ in $\mathfrak k$. Choose a positive subsystem $\Sigma^+$ and let $\mathfrak n = \sum_{\lambda\in\Sigma^+}\mathfrak g_\lambda$ (nilpotent Lie algebra). We get the Iwasawa decomposition $\mathfrak g =\mathfrak k+\mathfrak a+\mathfrak n$. Finally, $\mathfrak p_{min}=\mathfrak m+\mathfrak a+\mathfrak n$.

In general, $\mathfrak p = \mathfrak p_{min} + \sum_{\lambda\in\langle\Phi\rangle}\mathfrak g_{-\lambda}$, where $\Phi$ is a subset of of the system of simple roots of $\Sigma$.

It turns out $K$, as a subgroup of $G$, acts by left translations on $G/P$ and this action is transitive. In fact, the isotropy algebra at the basepoint is $\mathfrak k\cap\mathfrak p$, and if you count dimensions, you see that the orbit is open. It is also closed by compactness of $K$, so $K/K\cap P=G/P$. It remains to identify the left hand side with an orbit of the isotropy representation of the symmetric space $G/K$, namely, the action of $K$ on $\mathfrak s$, but this is not hard.

I can refer you to the nice book "An Introduction to Lie Groups and the Geometry of Homogeneous Spaces", by Andreas Arvanitoyeorgos (http://www.ams.org/bookstore?fn=20&arg1=stmlseries&ikey=STML-22).

Edit: Indeed $\mathfrak k\cap\mathfrak p =\mathfrak m + \sum_{\lambda\in\langle\Phi\rangle}(\mathfrak g_{\lambda}+\mathfrak g_{-\lambda})\cap\mathfrak k$ and you can select a point $x\in\mathfrak a\subset\mathfrak s$ such that $\lambda(x)=0$ if and only if $\lambda\in\langle\Phi\rangle$. The desired $K$-orbit is that through $x$.

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    $\begingroup$ I should add as a comment to this fairly old answer that the book Parabolic Geometries by Čap and Slovák, despite a slightly intimidating title, is excellent as an introduction to real and complex Lie groups and homogeneous spaces in general, and to answer my question in particular. (Arvanitoyeorgos's book is quite good, but only deals with the complex case of parabolic quotients.) $\endgroup$
    – Gro-Tsen
    Jul 21, 2014 at 0:31

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