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Equivariant cohomology of real flag manifolds
Let be a semisimple non-compact Riemannian symmetric space, where
and is the stabilizer of . Let be an orbit of
the (isotropy) representation of on ( is called a real flag
manifold). Let be the stabilizer of a maximal flat, totally
geodesic submanifold of which contains . We show that if all the simple
root multiplicities of are at least 2 then is connected and the
action of on is equivariantly formal. In the case when the
multiplicities are equal and at least 2, we will give a purely geometric proof
of a formula of Hsiang, Palais and Terng concerning . In particular,
this gives a conceptually new proof of Borel's formula for the cohomology ring
of an adjoint orbit of a compact Lie group.Comment: 11 pages, revised version (with corrections to the proofs of Lemma
2.2 and Theorem 1.1
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