Quantum cohomology of$G/P$and homology of affine Grassmannian

Thomas Lam Department of Mathematics, University of Michigan Mark Shimozono Department of Mathematics, Virginia Tech

TBD mathscidoc:1701.332012

Acta Mathematica, 204, (1), 49-90, 2008.2
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 We prove an unpublished result of D. Peterson which states that the quantum cohomology$QH$^{*}($G$/$P$) of a flag variety is, up to localization, a quotient of the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.
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@inproceedings{thomas2008quantum,
  title={Quantum cohomology of$G/P$and homology of affine Grassmannian},
  author={Thomas Lam, and Mark Shimozono},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203354714888721},
  booktitle={Acta Mathematica},
  volume={204},
  number={1},
  pages={49-90},
  year={2008},
}
Thomas Lam, and Mark Shimozono. Quantum cohomology of$G/P$and homology of affine Grassmannian. 2008. Vol. 204. In Acta Mathematica. pp.49-90. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203354714888721.
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