Schubert calculus on the grassmannian of hermitian lagrangian spaces

The grassmannian of hermitian lagrangian spaces in $$\mathbb{C}$^n\oplus $\mathbb{C}$^n$isanaturalcompactificationofthespaceofhermitian$n\times n$ matrices. We describe a Schubert-like, Whitney regular stratification on this space which has a Morse theoretic origin. We prove that these strata define closed subanalytic currents `{a} la R. Hardt, generating the integral homology of this space, we investigate their intersection theoretic properties, and we prove certain odd (in K-theoretic sense) Thom-Porteous type theorems.