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

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 for various symmetric pairs (G, K). We describe an interpretation of these formulas as representing the classes of particular types of degeneracy loci when evaluated at certain Chern classes. For the type A pair $$(SL(p+q,{\mathbb C}),S(GL(p,{\mathbb C}) \times GL(q,{\mathbb C})))$$ , such degeneracy loci are described explicitly, relative to a rank $$p+q$$ vector bundle V on a smooth complex variety X equipped with a flag of subbundles and a splitting of V as a direct sum of subbundles of ranks p and q. We conjecture similarly explicit descriptions of the degeneracy loci for all cases in types B and C.