TITLE: Hard Lefschetz Theorem for the Flag Manifolds
QUESTION [8 upvotes]: In the case of a generalized flag manifold $G/P$, we have an explicit description of their cohomology groups due to Borel.(See herehere for a description.) I would like to know what the hard Lefschetz theorem looks like in this presentation.

REPLY [13 votes]: To add to David's answer a bit:
The Borel description of $H(G/P)$ is the $W_P$-invariant subring of the $W$ coinvariants, $S^{W_P}_W$, where $W$ is the associated Weyl group and $W_P\subseteq W$ is the parabolic subgroup associated to $P$.  As David indicates above, we can choose a Schubert basis for $S^{W_P}_W$, and compute the matrices for the Lefschetz maps in that basis.  The Pieri rule allows us to interpret these Lefschetz matrices as so-called ''weighted path matrices'' with respect to the Bruhat order restricted to the set of maximal coset representatives of $W/W_P$.  A well known ''lemma'' of Gessel and Viennot is then helpful in computing these determinants (the original paper is available here).  These ideas are used  in this paper (now published in Journal of Algebra) to give a purely algebraic proof of the hard Lefschetz theorem for $S_W$ where $W$ is any finite Coxeter group, as well as for $S^{W_P}_W$ for certain maximal parabolics $W_P\subseteq W$.  One could probably also use these ideas to prove that $S^{W_P}_W$ is hard Lefschetz for any parabolic subgroup, although the Bruhat order for $W/W_P$ is more complicated.