Quantum cohomology of G/P and homology of affine Grassmannian

• 成果类型: Article
• 署名作者: Lam, Thomas; Shimozono, Mark
• 署名单位: University of Michigan System; University of Michigan; Virginia Polytechnic Institute & State University
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-010-0045-8
• 发表日期: 2010
• 页码: 49-90
• 关键词: gromov-witten invariants; Positivity

摘要

Let G be a simple and simply-connected complex algebraic group, P aS,aEuro parts per thousand 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 We prove an unpublished result of D. Peterson which states that the quantum cohomology QH (*)(G/P) of a flag variety is, up to localization, a quotient of the homology H (*)(Gr (G) )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.