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

Schubert calculus on the grassmannian of hermitian lagrangian spaces