Quantum cohomology of <Emphasis Type="Italic">G/P</Emphasis> and homology of affine Grassmannian

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 We prove an unpublished result of D. Peterson which states that the quantum cohomology QH ^*(G/P) of a flag variety is, up to localization, a quotient of the homology H _*(Gr _G ) of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.