The Riccati flow and singularities of Schubert varieties

Let $\operatorname{Gr}(m,n)$ be the Grassmannian of m-dimensional subspaces of an n-dimensional k-vector space, with $k = {\mathbf{R}}$ or C. Fix an $n \times n$ matrix R with coefficients in k. The Riccati Flow $\Phi$ on $\operatorname{Gr}(m,n)$ is the action of a one-parameter subgroup of ${\text{GL}_n}(k)$, given by ${\Phi _t}(\Lambda ) = {e^{Rt}}\Lambda$. We prove:

Theorem. Let X be a Schubert variety in $\operatorname{Gr}(m,n)$. Then there exists a Riccati flow $\Phi$ on X and a stable manifold W for $\Phi$ such that W is the smooth locus of X.

Corollary (over C). X as above is smooth if and only if the cohomology of X satisfies Poincaré Duality.