K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles splitting as direct sums

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 $G\mathrm{/}B$ for various symmetric pairs $(G,K)$. In type $A$, we realize the closures of $K=GL(p,\text{C})\times GL(q,\text{C})$-orbits on $GL(p+q,\text{C})\mathrm{/}B$ as universal degeneracy loci for a vector bundle over a variety which is equipped with a single flag of subbundles and which splits as a direct sum of subbundles of ranks $p$ and $q$. The precise description of such a degeneracy locus relies upon knowing a set-theoretic description of $K$-orbit closures, which we provide via a detailed combinatorial analysis of the poset of “$(p,q)$-clans,” which parametrize the orbit closures. We describe precisely how our formulas for the equivariant classes of $K$-orbit closures can be interpreted as formulas for the classes of such degeneracy loci in the Chern classes of the bundles involved. In the cases outside of type $A$, we suggest that the orbit closures should parametrize degeneracy loci involving a vector bundle equipped with a non-degenerate symmetric or skew-symmetric bilinear form, a single flag of subbundles which are isotropic or Lagrangian with respect to the form, and a splitting as a direct sum of subbundles with each summand satisfying some property (depending on $K$) with respect to the form. The precise description of such a degeneracy locus is conjectured for all cases in types $B$ and $C$.