Employing a simple and direct geometric approach, we prove formulas for a large class of degeneracy loci in types B, C, and D, including those coming from all isotropic Grassmannians. The results unify and generalize previous Pfaffian and determinantal formulas. Specializing to the Grassmannian case, we recover the remarkable theta- and eta-polynomials of Buch, Kresch, Tamvakis, and Wilson. Our method yields streamlined proofs which proceed in parallel for all four classical types, substantially simplifying previous work on the subject. In an appendix, we develop some foundational algebra and prove several Pfaffian identities. Another appendix establishes a basic formula for classes in quadric bundles.

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G=SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, calledK-k-Schur functions, whose highest-degree term is a k-Schur function. The dual basis in K-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations by means of Kashiwara’s thick affine flag manifold.

We discuss a natural action of the center of G on the Gromov–Witten numbers of G/B's and G/P's. This action is suggested by some problems in representation theory. The quantum Schubert calculus of Grassmannians is an easy consequence of this action. We also strengthen a theorem of Fulton and Woodward, in the case of Grasmannians.