Quantum cohomology of G/P and homology of affine Grassmannian

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 We prove an unpublished result of D. Peterson which states that the quantum cohomology QH * (G/P ) of a flag variety is, up to localization, a quotient of the homology H * (Gr G ) of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 the 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.

k-Schur Functions and Affine Schubert Calculus

“…The symmetry of affine Stanley symmetric functions follows from the commutativity of Peterson's subalgebra, and the positivity in terms of affine Schur functions is established via the relationship between affine Schubert calculus and quantum Schubert calculus [88,116]. The affine-quantum connection was also discovered by Peterson.…”

“…Find general formulae for j w in terms of A x . See [88] for a formula in terms of quantum Schubert polynomials, which however is not very explicit.…”

“…Find a combinatorial formula for the j-basis. It follows from work of Peterson (see also [79,88]) that the coefficients j x w are positive.…”

“…In particular Peterson's j-basis (see Chapter 4, Section 4.5), which is defined algebraically using a leading term condition for an expansion in the divided difference basis, has an analogue (called the k-basis in [85]) that corresponds to ideal sheaves of Schubert varieties in the affine Grassmannian. Peterson's "quantum equals affine" theorem [131,88] (see Chapter 4, Section 4.7) has an analogue in K-theory: the structure sheaves of opposite Schubert varieties in the quantum K-theory QK T (G/B) of finite-dimensional flag varieties G/B, appear to multiply in the same way as the structure sheaves of the Schubert varieties in the K-homology K T (Gr G ) of the affine Grassmannian [83]. To establish this connection one must prove a conjectural Chevalley formula of Lenart and Postnikov [114,Conjecture 17.1] for quantum K-theory.…”

Schubert polynomials for the affine Grassmannian of the symplectic group

“…Using the nonequivariant case ℓ(y) = ℓ(x) of (4.12), these constants are given by the coefficients j y x of (4.11). But these are known to be nonnegative from the work of Peterson [22] and Lam and Shimozono [17]; they are equal to certain three-point genus zero Gromov-Witten invariants of the (finite) flag variety.…”