In this talk, I will report on the following joint work with Nantel Bergeron of York University:
We give the formula for the multiplication of an arbitrary Schubert class in the cohomology of a symplectic flag manifold by a special Schubert class pulled back from the Lagrangian Grassmannian. This formula is expressed in both terms of chains in the Bruhat order, and in terms of the cycle structure a certain permutations, showing it to be a common generalization of the Pieri-type formula for the Lagrangian Grassmannian and that for the ordinary flag manifold. Our proof uses results on the Bruhat order, identities of structure constants and intersections of Schubert varieties.