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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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