Quantum cohomology of G/P and homology of affine Grassmannian

  title={Quantum cohomology of G/P and homology of affine Grassmannian},
  author={Thomas Lam and Mark Shimozono},
  journal={Acta Mathematica},
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 We prove an unpublished result of D. Peterson which states that the quantum cohomology QH*(G/P) of a flag variety is, up to localization, a quotient of the homology H*(GrG) of the affine Grassmannian GrG 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*(GrG), establishing the equivalence of the quantum and… 
Peterson-Lam-Shimozono's theorem is an affine analogue of quantum Chevalley formula
We give a new proof of an unpublished result of Peterson, proved by Lam-Shimozono, which identifies explicitly the structure constants for the multiplications of Schubert classes in the T
Affine nil-Hecke algebras and Quantum cohomology
. Let G be a compact, connected Lie group and T ⊂ G a maximal torus. Let ( M, ω ) be a monotone closed symplectic manifold equipped with a Hamiltonian action of G . We construct a module action of
Applying parabolic Peterson: affine algebras and the quantum cohomology of the Grassmannian
The Peterson isomorphism relates the homology of the affine Grassmannian to the quantum cohomology of any flag variety. In the case of a partial flag, Peterson's map is only a surjection, and one
Total positivity, Schubert positivity, and Geometric Satake
Maximal Newton Points and the Quantum Bruhat Graph
We discuss a surprising relationship between the partially ordered set of Newton points associated to an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem
Quantum characteristic classes, moment correspondences and the Hamiltonian groups of coadjoint orbits
For any coadjoint orbit G/L, we determine all useful terms of the associated SavelyevSeidel morphism defined on H−∗(ΩG). Immediate consequences are: (1) the dimension of the kernel of the natural map
Applications of the theory of Floer to symmetric spaces
We quantize the problem considered by Bott-Samelson who applied Morse theory to compact symmetric spaces G/K and the associated real flag manifolds Λ. The starting point is the construction of a
Peterson Isomorphism in $K$-theory and Relativistic Toda Lattice
The $K$-homology ring of the affine Grassmannian of $SL_n(C)$ was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On
Schubert polynomials for the affine Grassmannian of the symplectic group
We study the Schubert calculus of the affine Grassmannian Gr of the symplectic group. The integral homology and cohomology rings of Gr are identified with dual Hopf algebras of symmetric functions,
Maximal Newton polygons via the quantum Bruhat graph
This paper discusses a surprising relationship between the quantum cohomology of the variety of complete flags and the partially ordered set of Newton polygons associated to an element in the affine


Polynomial Representatives of Schubert Classes in Qh
We show how the quantum Chevalley formula for G/B, as stated by Peterson and proved rigorously by Fulton and Woodward, combined with ideas of Fomin, S. Gelfand and Postnikov, leads to a formula which
Schubert polynomials for the affine Grassmannian
Let G be a complex simply connected simple group and K a maximal compact subgroup. Let F = C((t)) denote the field of formal Laurent series and O = C[[t]] the ring of formal power series. The
We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence,
Quantum Schubert polynomials
where In is the ideal generated by symmetric polynomials in x1,... ,xn without constant term. Another, geometric, description of the cohomology ring of the flag manifold is based on the decomposition
Quantum Bruhat graph and Schubert polynomials
The quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of G/B. We enhance a result of
The nil Hecke ring and cohomology of G/P for a Kac-Moody group G.
A ring R is constructed, which is very simply and explicitly defined as a functor of W together with the W-module [unk] alone and such that all these four structures on H(*)(G/B) arise naturally from the ring R.
Quantum cohomology of flag manifolds G/B and quantum Toda lattices
Let G be a connected semi-simple complex Lie group, B its Borel subgroup, T a maximal complex torus contained in B, and Lie (T ) its Lie algebra. This setup gives rise to two constructions; the
Affine Insertion and Pieri Rules for the Affine Grassmannian
We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n,C). Our main results are: 1) Pieri rules for the Schubert bases of H^*(Gr) and H_*(Gr),
Mixed Bruhat Operators and Yang-Baxter Equations for Weyl Groups
We introduce and study a family of operators which act in the group algebra of a Weyl group W and provide a multiparameter solution to the quantum Yang-Baxter equations of the corresponding type.
Equivariant quantum cohomology of homogeneous spaces.
We prove a Chevalley formula for the equivariant quantum multiplication of two Schubert classes in the homogeneous variety X=G/P. As in the case when X is a Grassmannian, studied by the author in a