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

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