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 We prove an unpublished result of D. Peterson which states that the quantum cohomology QH *(G/P) of a flag variety is, up to localization, a quotient of the homology H *(Gr G ) of the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.
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References
Bertram, A., Quantum Schubert calculus. Adv. Math., 128 (1997), 289–305.
Bezrukavnikov, R., Finkelberg, M. & Mirković, I., Equivariant homology and Ktheory of affine Grassmannians and Toda lattices. Compos. Math., 141 (2005), 746–768.
Bott, R., The space of loops on a Lie group. Michigan Math. J., 5 (1958), 35–61.
Bourbaki, N., Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. Actualités Scientifiques et Industrielles, 1337. Hermann, Paris, 1968.
Brenti, F., Fomin, S. & Postnikov, A., Mixed Bruhat operators and Yang–Baxter equations for Weyl groups. Int. Math. Res. Not., 8 (1999), 419–441.
Buch, A. S., Kresch, A. & Tamvakis, H., Gromov–Witten invariants on Grassmannians. J. Amer. Math. Soc., 16 (2003), 901–915.
Chaput, P. E., Manivel, L. & Perrin, N., Quantum cohomology of minuscule homogeneous spaces. II. Hidden symmetries. Int. Math. Res. Not. IMRN, 22 (2007), Art. ID rnm107.
Dyer, M. J., Hecke algebras and shellings of Bruhat intervals. Compos. Math., 89 (1993), 91–115.
Fomin, S., Gelfand, S. & Postnikov, A., Quantum Schubert polynomials. J. Amer. Math. Soc., 10 (1997), 565–596.
Fulton, W. & Woodward, C., On the quantum product of Schubert classes. J. Algebraic Geom., 13 (2004), 641–661.
Garland, H. & Raghunathan, M. S., A Bruhat decomposition for the loop space of a compact group: a new approach to results of Bott. Proc. Nat. Acad. Sci. U.S.A., 72:12 (1975), 4716–4717.
Ginzburg, V., Perverse sheaves on a Loop group and Langlands’ duality. Preprint, 1995. arXiv:alg-geom/9511007.
Humphreys, J. E., Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, 9. Springer, New York, 1978.
Kac, V. G., Infinite-dimensional Lie Algebras. Cambridge University Press, Cambridge, 1990.
Kim, B., Quantum cohomology of flag manifolds G/B and quantum Toda lattices. Ann. of Math., 149 (1999), 129–148.
Kostant, B., Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight ϱ. Selecta Math., 2 (1996), 43–91.
Kostant, B. & Kumar, S., The nil Hecke ring and cohomology of G/P for a Kac–Moody group G. Adv. Math., 62 (1986), 187–237.
Kumar, S., Kac–Moody Groups, their Flag Varieties and Representation Theory. Progress in Mathematics, 204. Birkhäuser, Boston, MA, 2002.
Lam, T., Schubert polynomials for the affine Grassmannian. J. Amer. Math. Soc., 21 (2008), 259–281.
Lam, T., Lapointe, L., Morse, J. & Shimozono, M., Affine insertion and Pieri rules for the affine Grassmannian. To appear in Mem. Amer. Math. Soc.
Lam, T. & Shimozono, M., Dual graded graphs for Kac–Moody algebras. Algebra Number Theory, 1 (2007), 451–488.
Lapointe, L., Lascoux, A. & Morse, J., Tableau atoms and a new Macdonald positivity conjecture. Duke Math. J., 116 (2003), 103–146.
Lapointe, L. & Morse, J., Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions. J. Combin. Theory Ser. A, 112 (2005), 44–81.
_____ Quantum cohomology and the k-Schur basis. Trans. Amer. Math. Soc., 360:4 (2008), 2021–2040.
Mare, A. L., Polynomial representatives of Schubert classes in QH*(G/B). Math. Res. Lett., 9 (2002), 757–769.
Mihalcea, L. C., Positivity in equivariant quantum Schubert calculus. Amer. J. Math., 128 (2006), 787–803.
_____ On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms. Duke Math. J., 140 (2007), 321–350.
Mitchell, S.A., Quillen’s theorem on buildings and loop groups. Enseign. Math., 34 (1988), 123–166.
Peterson, D., Quantum cohomology of G/P. Lecture notes, Massachusetts Institute of Technology, Cambridge, MA, Spring 1997.
Postnikov, A., Quantum Bruhat graph and Schubert polynomials. Proc. Amer. Math. Soc., 133 (2005), 699–709.
Rietsch, K., Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties. J. Amer. Math. Soc., 16 (2003), 363–392.
Spanier, E. H., Algebraic Topology. Springer, New York, 1981.
Verma, D.-N., Möbius inversion for the Bruhat ordering on a Weyl group. Ann. Sci. École Norm. Sup., 4 (1971), 393–398.
Woodward, C. T., On D. Peterson’s comparison formula for Gromov–Witten invariants of G/P. Proc. Amer. Math. Soc., 133:6 (2005), 1601–1609.
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This work was partially supported by NSF grants DMS-0401012, DMS-0600677, DMS-0652641 and DMS-0652648.
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Lam, T., Shimozono, M. Quantum cohomology of G/P and homology of affine Grassmannian. Acta Math 204, 49–90 (2010). https://doi.org/10.1007/s11511-010-0045-8
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DOI: https://doi.org/10.1007/s11511-010-0045-8