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Quantum cohomology of $$G/P$$ and homology of affine Grassmannian. (English) Zbl 1216.14052
Let $$G$$ be a simple and simply connected complex algebraic group, $$\text{Gr}_G$$ be its affine Grassmanian, and $$P\subset G$$ a parabolic subgroup. The authors prove that the quantum cohomology ring of a flag manifold $$QH^*(G/P)$$ is a quotient of $$H^*(\text{Gr}_G)$$ after localization, and give the quotient map explicitly in terms of Schubert classes. This result was stated without a proof by Dale Peterson in 1997.
The authors’ proof also extends to the equivariant setting. Partial manifestations of this correspondence for $$G/B$$, where $$B$$ is the Borel subgroup, appear in the work of R. Bezrukavnikov, M. Finkelberg and I. Mirković [Compos. Math. 141, No. 3, 746–768 (2005; Zbl 1065.19004)] and B. Kim [Ann. Math. (2) 149, No. 1, 129–148 (1999; Zbl 1054.14533)]. But even for $$P=B$$, the ring property of the quotient map is new. For $$G=SL_{k+1}(\mathbb{C})$$, a closely related ring homomorphism was studied by L. Lapointe and J. Morse [J. Comb. Theory, Ser. A 112, No. 1, 44–81 (2005; Zbl 1120.05093)] in terms of $$k$$-Schur functions. The authors note that it looks promising to also compare other structures, such as mirror symmetry on $$QH^*(G/P)$$ and Hopf algebra structure with nil-Hecke action on $$H^*(\text{Gr}_G)$$.
For $$P=B$$, the proof relies on the replationship between the quantum Bruhat graph and the Bruhat order on the elements of the affine Weyl group with a large translation component. It also utilizes algebraic properties of $$QH^*(G/B)$$ including the $$T$$-equivariant quantum Chevalley formula of Peterson, proved by L. C. Mihalcea [Duke Math. J. 140, No. 2, 321–350 (2007; Zbl 1135.14042)] ($$T\subset G$$ is a maximal torus). A byproduct of the proof are expressions for affine Schubert classes in terms of generating functions over paths in the quantum Bruhat graph.
For $$P\neq B$$, the authors utilize the Coxeter combinatorics of the affinization of the Weyl group of the Levy factor of $$P$$. It allows them to use the comparison formula of Woodward to relate quantum Chevalley formulas of $$QH^*(G/P)$$ and $$QH^*(G/B)$$. A formula for quantum multiplication by a Schubert class labeled by the reflection in the highest root is then deduced. It turns out that the ring homomorphism of Lapointe and Morse differs from the one here by the strange duality of $$QH^*(G/P)$$ discovered by P.-E. Chaput, L. Manivel and N. Perrin [Int. Math. Res. Not. 2007, No. 22, Article ID rnm107, 29 p. (2007; Zbl 1142.14033)].

##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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##### References:
 [1] Bertram, A., Quantum Schubert calculus. Adv. Math., 128 (1997), 289–305. · Zbl 0945.14031 · doi:10.1006/aima.1997.1627 [2] Bezrukavnikov, R., Finkelberg, M. & Mirković, I., Equivariant homology and Ktheory of affine Grassmannians and Toda lattices. Compos. Math., 141 (2005), 746–768. · Zbl 1065.19004 · doi:10.1112/S0010437X04001228 [3] Bott, R., The space of loops on a Lie group. Michigan Math. J., 5 (1958), 35–61. · Zbl 0096.17701 · doi:10.1307/mmj/1028998010 [4] 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. · Zbl 0186.33001 [5] Brenti, F., Fomin, S. & Postnikov, A., Mixed Bruhat operators and Yang–Baxter equations for Weyl groups. Int. Math. Res. Not., 8 (1999), 419–441. · Zbl 0978.22008 · doi:10.1155/S1073792899000215 [6] Buch, A. S., Kresch, A. & Tamvakis, H., Gromov–Witten invariants on Grassmannians. J. Amer. Math. Soc., 16 (2003), 901–915. · Zbl 1063.53090 · doi:10.1090/S0894-0347-03-00429-6 [7] 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. · Zbl 1142.14033 [8] Dyer, M. J., Hecke algebras and shellings of Bruhat intervals. Compos. Math., 89 (1993), 91–115. · Zbl 0817.20045 [9] Fomin, S., Gelfand, S. & Postnikov, A., Quantum Schubert polynomials. J. Amer. Math. Soc., 10 (1997), 565–596. · Zbl 0912.14018 · doi:10.1090/S0894-0347-97-00237-3 [10] Fulton, W. & Woodward, C., On the quantum product of Schubert classes. J. Algebraic Geom., 13 (2004), 641–661. · Zbl 1081.14076 [11] 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. · Zbl 0344.55009 [12] Ginzburg, V., Perverse sheaves on a Loop group and Langlands’ duality. Preprint, 1995. arXiv:alg-geom/9511007. [13] Humphreys, J. E., Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, 9. Springer, New York, 1978. · Zbl 0447.17001 [14] Kac, V. G., Infinite-dimensional Lie Algebras. Cambridge University Press, Cambridge, 1990. · Zbl 0716.17022 [15] Kim, B., Quantum cohomology of flag manifolds G/B and quantum Toda lattices. Ann. of Math., 149 (1999), 129–148. · Zbl 1054.14533 · doi:10.2307/121021 [16] Kostant, B., Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight . Selecta Math., 2 (1996), 43–91. · Zbl 0868.14024 · doi:10.1007/BF01587939 [17] 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. · Zbl 0641.17008 · doi:10.1016/0001-8708(86)90101-5 [18] Kumar, S., Kac–Moody Groups, their Flag Varieties and Representation Theory. Progress in Mathematics, 204. Birkhäuser, Boston, MA, 2002. · Zbl 1026.17030 [19] Lam, T., Schubert polynomials for the affine Grassmannian. J. Amer. Math. Soc., 21 (2008), 259–281. · Zbl 1149.05045 · doi:10.1090/S0894-0347-06-00553-4 [20] Lam, T., Lapointe, L., Morse, J. & Shimozono, M., Affine insertion and Pieri rules for the affine Grassmannian. To appear in Mem. Amer. Math. Soc. · Zbl 1208.14002 [21] Lam, T. & Shimozono, M., Dual graded graphs for Kac–Moody algebras. Algebra Number Theory, 1 (2007), 451–488. · Zbl 1200.05249 · doi:10.2140/ant.2007.1.451 [22] Lapointe, L., Lascoux, A. & Morse, J., Tableau atoms and a new Macdonald positivity conjecture. Duke Math. J., 116 (2003), 103–146. · Zbl 1020.05069 · doi:10.1215/S0012-7094-03-11614-2 [23] 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. · Zbl 1120.05093 · doi:10.1016/j.jcta.2005.01.003 [24] _____ Quantum cohomology and the k-Schur basis. Trans. Amer. Math. Soc., 360:4 (2008), 2021–2040. · Zbl 1132.05060 · doi:10.1090/S0002-9947-07-04287-0 [25] Mare, A. L., Polynomial representatives of Schubert classes in QH*(G/B). Math. Res. Lett., 9 (2002), 757–769. · Zbl 1054.14069 [26] Mihalcea, L. C., Positivity in equivariant quantum Schubert calculus. Amer. J. Math., 128 (2006), 787–803. · Zbl 1099.14047 · doi:10.1353/ajm.2006.0026 [27] _____ On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms. Duke Math. J., 140 (2007), 321–350. · Zbl 1135.14042 · doi:10.1215/S0012-7094-07-14024-9 [28] Mitchell, S.A., Quillen’s theorem on buildings and loop groups. Enseign. Math., 34 (1988), 123–166. · Zbl 0704.55006 [29] Peterson, D., Quantum cohomology of G/P. Lecture notes, Massachusetts Institute of Technology, Cambridge, MA, Spring 1997. [30] Postnikov, A., Quantum Bruhat graph and Schubert polynomials. Proc. Amer. Math. Soc., 133 (2005), 699–709. · Zbl 1051.05078 · doi:10.1090/S0002-9939-04-07614-2 [31] Rietsch, K., Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties. J. Amer. Math. Soc., 16 (2003), 363–392. · Zbl 1057.14065 · doi:10.1090/S0894-0347-02-00412-5 [32] Spanier, E. H., Algebraic Topology. Springer, New York, 1981. [33] Verma, D.-N., Möbius inversion for the Bruhat ordering on a Weyl group. Ann. Sci. École Norm. Sup., 4 (1971), 393–398. · Zbl 0236.20035 [34] 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. · Zbl 1077.14085 · doi:10.1090/S0002-9939-05-07709-9
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