Schubert calculus on the grassmannian of hermitian lagrangian spaces

TL;DR

This paper develops a Schubert calculus framework on the Grassmannian of Hermitian Lagrangian spaces, revealing its topology, stratification, and intersection properties with applications to Morse theory and K-theoretic Thom-Porteous theorems.

Contribution

It introduces a new Schubert-like stratification on the Hermitian Lagrangian Grassmannian with Morse theoretic and intersection-theoretic analysis, including novel Thom-Porteous type results.

Findings

Stratification generates the integral homology of the space.

Strata define closed subanalytic currents with Morse theoretic origin.

Proves odd K-theoretic Thom-Porteous theorems.

Abstract

The grassmannian of hermitian lagrangian spaces in $\mathbb{C}^n\oplus \mathbb{C}^n$ is a natural compactification of the space of hermitian $n\times n$ 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.