Equivariant cohomology of real flag manifolds

Abstract

Let $P=G/K$ be a semisimple non-compact Riemannian symmetric space, where $G=I_0(P)$ and $K=G_p$ is the stabilizer of $p\in P$. Let $X$ be an orbit of the (isotropy) representation of $K$ on $T_p(P)$ ($X$ is called a real flag manifold). Let $K_0\subset K$ 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 $G/K$ are at least 2 then $K_0$ is connected and the action of $K_0$ 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 $H^*(X)$. 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.