Quantum cohomology of G/P and homology of affine Grassmannian

Abstract

Let G be a simple and simply-connected complex algebraic group, P ⊂ G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization, a quotient of the homology H_*(Gr_G) of the affine Grassmannian Gr_G of G. As a consequence, all three-point genus-zero Gromov–Witten invariants of G/P are identified with homology Schubert structure constants of H_*(Gr_G), establishing the equivalence of the quantum and homology affine Schubert calculi.

For the case G = B, we use Mihalcea’s equivariant quantum Chevalley formula for QH^*(G/B), together with relationships between the quantum Bruhat graph of Brenti, Fomin and Postnikov and the Bruhat order on the affine Weyl group. As byproducts we obtain formulae for affine Schubert homology classes in terms of quantum Schubert polynomials. We give some applications in quantum cohomology.

Our main results extend to the torus-equivariant setting.