×

Equivariant \(K\)-theory of affine flag manifolds and affine Grothendieck polynomials. (English) Zbl 1173.19004

Let \(G\) be an affine Lie group, \(B\) a Borel subgroup and \(X= G/B\) the corresponding affine flag manifold which is decomposed into the union of affine Schubert varieties \(X_w = \overline{BwB/B}\). \(X\) is an infinite-dimensional (not quasi-compact) scheme over \(\mathbb C\) and its \(B\)-orbits are parametrized by the elements of the Weyl group \(W\). Each \(B\) orbit is a locally closed subscheme with finite codimension and is isomorphic to the scheme \(\mathbb A^\infty = \text{Spec} (\mathbb C(x_1,x_2,\dots,))\). The \(K\)-group is decomposed into the product \(K_B(X) \cong \prod_{w\in W} K_B(pt)[\mathcal{O}_{X^w}]\). The group \(K_B(pt)\) is isomorphic to the group ring \(\mathbb Z[P]\) of the weight lattice \(P\) of a maximal torus of \(B\). Similar to the finite-dimensional case there is a homomorphism \(\mathbb Z[P] \otimes \mathbb Z[P] \cong K_B(pt) \otimes K_B(pt) \to K_B(X)\) which factors through the equivariant Atiyah-Hirzebruch homomorphism \(\mathbb Z[P] \bigotimes_{\mathbb Z[P]^W} \mathbb Z[P] \to K_B(X)\), where \(\mathbb Z[P]^W\) is the ring of invariants with respect to the action of the Weyl group \(W\).
In the affine case, this morphism in injective but not surjective, not all \([{\mathcal O}_{X^w}]\) are in the image of this morphism. Nevertheless, after localization by a generator \(\delta\) of null roots, taking tensor product with the subring \(\mathbb Q[\delta]\) we get the main result of the paper: Theorem 4.4: all the elements \([{\mathcal O}_{X^w}]\) are in \(\mathbb Q[\delta] \bigotimes_{\mathbb Z[e^{\pm\delta}]} K_B(X)\). The authors call the elements of \(R \bigotimes_{\mathbb Z[e^{\pm\delta}]} \mathbb Z[P] \bigotimes_{\mathbb Z[P]^W} \mathbb Z[P]\) the affine Grothendieck polynomials.

MSC:

19L47 Equivariant \(K\)-theory
14M17 Homogeneous spaces and generalizations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] R. Bott, The space of loops on a Lie group , Michigan Math. J. 5 (1958), 35–61. · Zbl 0096.17701
[2] A. S. Buch, A Littlewood-Richardson rule for the \(K\)-theory of Grassmannians , Acta Math. 189 (2002), 37–78. · Zbl 1090.14015
[3] M. Demazure, “Désingularisation des variétés de Schubert généralisées” in Collection of Articles Dedicated to Henri Cartan on the Occasion of his 70th Birthday, I , Ann. Sci. École Norm. Sup. (4) 7 , Gauthier-Villars, Paris, 1974, 53–88. · Zbl 0312.14009
[4] S. Fomin and A. N. Kirillov, “The Yang-Baxter equation, symmetric functions, and Schubert polynomials” in Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, Italy, 1993) , Discrete Math. 153 , Elsevier, Amsterdam, 1996, 123–143. · Zbl 0852.05078
[5] W. Fulton and A. Lascoux, A Pieri formula in the Grothendieck ring of a flag bundle , Duke Math. J. 76 (1994), 711–729. · Zbl 0840.14007
[6] R. Hartshorne, Residues and Duality , with appendix by P. Deligne, Lecture Notes in Math. 20 , Springer, Berlin, 1966. · Zbl 0212.26101
[7] M. Kashiwara, “The flag manifold of Kac-Moody Lie algebra” in Algebraic Analysis, Geometry, and Number Theory (Baltimore, Md., 1988) , Johns Hopkins Univ. Press, Baltimore, Md., 1989, 161–190. · Zbl 0764.17019
[8] -, “Kazhdan-Lusztig conjecture for a symmetrizable Kac-Moody Lie algebra” in The Grothendieck Festschrift, Vol. II , Progr. Math. 87 , Birkhäuser, Boston, 1990, 407–433. · Zbl 0727.17013
[9] M. Kashiwara and P. Schapira, Sheaves on Manifolds , Grundlehren Math. Wiss. 292 , Springer, Berlin, 1990. · Zbl 0709.18001
[10] -, Categories and Sheaves , Grundlehren Math. Wiss. 332 , Springer, Berlin, 2006. · Zbl 1118.18001
[11] A. Knutson and T. Tao, Puzzles and (equivariant) cohomology of Grassmannians , Duke Math. J. 119 (2003), 221–260. · Zbl 1064.14063
[12] M. Kogan, Generalization of Schensted insertion algorithm to the cases of hooks and semi-shuffles , J. Combin. Theory Ser. A 102 (2003), 110–135. · Zbl 1019.05066
[13] B. Kostant and S. Kumar, \(T\)-equivariant \(K\)-theory of generalized flag varieties , J. Differential Geom. 32 (1990), 549–603. · Zbl 0731.55005
[14] S. Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory , Progr. Math. 204 , Birkhäuser, Boston, 2002. · Zbl 1026.17030
[15] T. Lam, Affine Stanley symmetric functions , Amer. J. Math. 128 (2006), 1553–1586. · Zbl 1107.05095
[16] -, Schubert polynomials for the affine Grassmannian , J. Amer. Math. Soc. 21 (2008), 259–281. · Zbl 1149.05045
[17] T. Lam, L. Lapointe, J. Morse, and M. Shimozono, Affine insertion and Pieri rules for the affine Grassmannian , preprint,\arxivmath/0609110v3[math.CO] · Zbl 1208.14002
[18] L. Lapointe, A. Lascoux, and J. Morse, Tableau atoms and a new Macdonald positivity conjecture , Duke Math. J. 116 (2003), 103–146. · Zbl 1020.05069
[19] L. Lapointe and J. Morse, Quantum cohomology and the \(k\)-Schur basis , Trans. Amer. Math. Soc. 360 , no. 4 (2008), 2021–2040. · Zbl 1132.05060
[20] A. Lascoux and M.-P. SchüTzenberger, Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux , C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), 629–633. · Zbl 0542.14030
[21] C. Lenart and F. Sottile, A Pieri-type formula for the K-theory of a flag manifold , Trans. Amer. Math. Soc. 359 , no. 5 (2007), 2317–2342. · Zbl 1111.14047
[22] P. Magyar, Notes on Schubert classes of a loop group , preprint, \arxiv0705.3826v1[math.RT]
[23] P. Pragacz, “Algebro-geometric applications of Schur \(S\)- and \(Q\)-polynomials” in Topics in Invariant Theory (Paris, 1989/1990) , Lecture Notes in Math. 1478 , Springer, Berlin, 1991, 130–191. · Zbl 0783.14031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.