# Flag manifolds (=R-spaces): quotients by parabolic subgroups vs. isotropy representation

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, and preferably with additional details on the connection between flag manifolds and Riemann symmetric spaces (e.g., on when one is the other, and conversely)?