K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear form

TL;DR

This paper derives explicit formulas for the cohomology classes of K-orbit closures on flag varieties, interpreting them as universal degeneracy loci for vector bundles with symmetric or skew-symmetric forms, using equivariant localization.

Contribution

It provides new equivariant cohomology formulas for K-orbit closures as universal degeneracy loci, connecting geometric orbit structures with bundle characteristic classes.

Findings

Formulas for torus-equivariant classes of K-orbit closures

Interpretation of orbit closures as degeneracy loci

Connection between orbit classes and Chern classes

Abstract

We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$, where $G = GL(n,\C)$, and where $K$ is one of the symmetric subgroups $O(n,\C)$ or $Sp(n,\C)$. We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formulas for the classes of such loci in terms of the Chern classes of the various bundles.