Equivariant cohomology of real flag manifolds

Let P=G/KP=G/K be a semisimple non-compact Riemannian symmetric space, where G=I0(P)G=I0(P) and K=GpK=Gp is the stabilizer of p∈Pp∈P. Let X be an orbit of the (isotropy) representation of K   on Tp(P)Tp(P) (X   is called a real flag manifold). Let K0⊂KK0⊂K be the stabilizer of a maximal flat, totally geodesic submanifold of P which contains p  . We show that if all the simple root multiplicities of G/KG/K are at least 2 then K0K0 is connected and the action of K0K0 on X   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 H∗(X)H∗(X). 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.