A Pieri-type formula for isotropic flag manifolds

We give the formula for the multiplication of a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from a Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type B (respectively, type C) Schubert polynomial by the Schur P-polynomial p_m (respectively, the Schur Q-polynomial q_m). Geometric constructions and intermediate results allow us to ultimately deduce this formula from multiplication formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the Coxeter group B_n, identities of the structure constants for the Schubert basis of cohomology, and intersections of Schubert varieties. We show these identities follow from the Pieri-type formula, except some `hidden symmetries' of the structure constants. Our analysis leads to a new partial order on the Coxeter group B_n and formulas for many of these structure constants.