Schubert calculus on the Grassmannian of hermitian lagrangian spaces

With an eye towards index theoretic applications we describe a Schubert like stratification on the Grassmannian of hermitian lagrangian spaces in Cn⊕Cn. This is a natural compactification of the space of hermitian n×n matrices. The closures of the strata define integral cycles, and we investigate their intersection theoretic properties. We achieve this by blending Morse theoretic ideas, with techniques from o-minimal (or tame) geometry and geometric integration theory.