Equivariant \(K\)-Chevalley rules for Kac-Moody flag manifolds.

*(English)*Zbl 1328.19012The paper under review supplies \(K\)-theoretic versions of the equivariant Chevalley rule holding for the infinite dimensional spaces known as Kac-Moody thick flag manifolds, investigated in great detail by M. Kashiwara [in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 161–190 (1989; Zbl 0764.17019)], in the realm of infinite dimensional Lie algebras. Within this framework, the authors prove four cancellation-free Chevalley formulas (Theorem 3.4 and Theorem 4.8), i.e. expansions with all positive coefficients of the product of the equivariant class of an equivariant line bundle with an arbitrary Schubert variety.

In the finite-dimensional context, recall that if \(G\) is a semisimple Lie group and \(B\) a Borel subgroup, the homogeneous space \(G/B\) is a projective variety, because \(B\) is parabolic (indeed is the minimal parabolic subgroup of \(G\)). The stratification of \(G/B\) in terms of closures of affine cells of a natural cellular decomposition was already studied by C. Chevalley [Proc. Symp. Pure Math. 56, 1–23 (1994; Zbl 0824.14042)]. For special choices of \(G\) and \(B\) one recovers more classical situations. Two main instances are provided by the complete flags of \({\mathbb C}^n\) and the Grassmannian \(G(k,n)\), parametrizing inclusions \(0\subseteq W\subseteq {\mathbb C}^n\), where \(W\) is a vector subspace of \({\mathbb C}^n\) of dimension \(k\). The Chevalley formula for \(G(k,n)\) rules precisely the intersection \(\sigma_1\sigma_\lambda\) of the generator of the Picard group of \(G(k, n)\) with arbitrary Schubert cycles \(\sigma_\lambda\) (parametrized by partitions contained in a \(k\times (n-k)\) rectangle). For the manifold \(Fl({\mathbb C}^n)\) parametrizing complete flags of subspaces of \({\mathbb C}^n\), the Chevalley rule is known as Monk rule. It prescribes the intersection of the basis of the Picard group, generated by the \(n\) classes corresponding to simple permutations, with natural defined Schubert varieties associated to some fixed reference flag. Equivariant versions of the Chevalley formula for Grassmannians are well known (see e.g. A. Knutson and T. Tao [Duke Math. J. 119, No. 2, 221–260 (2003; Zbl 1064.14063)]).

The transition to the equivariant setting is not straightforward but it naturally demands to be considered and investigated. In fact, one can think of the equivariant cohomology, and thus of the corresponding equivariant K-theory, of \(G/B\) with respect to the action of a maximal sub-torus of \(B\). Equivariant Chevalley formulas in the \(K\)-theory of \(G/B\) are also well known and yield the so-called Monk’s formula in type A. So, the generalization to the Kac-Moody thick flag varieties, dealt with in the paper under review, is a truly important step, and it can be viewed as as a natural prosecution of previous investigations published in a couple of articles by the first author and A. Postnikov [Int. Math. Res. Not. 2007, No. 12, Article ID rnm038, 65 p. (2007; Zbl 1137.14037)] and [Trans. Am. Math. Soc. 360, No. 8, 4349–4381 (2008; Zbl 1211.17021)].

It is remarkable that the formulas the authors obtain specialize to those for the classical flag manifolds recalled above. This desirable feature enables to detect and to fix a gap in the proof of previous results by S. Griffeth and A. Ram [Eur. J. Comb. 25, No. 8, 1263–1283 (2004; Zbl 1076.14068)] and H. Pittie and A. Ram [Electron. Res. Announc. Am. Math. Soc. 5, No. 14, 102–107 (1999; Zbl 0947.14025)].

The main tools used in the paper is the theory of Lakshmibay-Seshadri paths, treated in Section 3.4, as well as the alcove model, described in the aforementioned papers by Lenart and Postnikov, and many more technicalities which are partly explained and partly precisely referred to the appropriate literature.

The authors devote the final Section 5 of the article to describe a few illuminating examples, including the very useful one concerning the affine Grassmannian. One should dutifully add that the various Kac-Moody Chevalley rules have been implemented in the software sage which will be soon available for free public distribution. The paper ends with an essential bibliographical list. The references, on the other hand, are so precisely distributed along the exposition that the interested reader can find, although not without a personal effort, his/her own path to orient him/herself into the labyrinth of the many sophisticated mathematical techniques and knowledges, invoked and displayed in this truly fascinating article.

In the finite-dimensional context, recall that if \(G\) is a semisimple Lie group and \(B\) a Borel subgroup, the homogeneous space \(G/B\) is a projective variety, because \(B\) is parabolic (indeed is the minimal parabolic subgroup of \(G\)). The stratification of \(G/B\) in terms of closures of affine cells of a natural cellular decomposition was already studied by C. Chevalley [Proc. Symp. Pure Math. 56, 1–23 (1994; Zbl 0824.14042)]. For special choices of \(G\) and \(B\) one recovers more classical situations. Two main instances are provided by the complete flags of \({\mathbb C}^n\) and the Grassmannian \(G(k,n)\), parametrizing inclusions \(0\subseteq W\subseteq {\mathbb C}^n\), where \(W\) is a vector subspace of \({\mathbb C}^n\) of dimension \(k\). The Chevalley formula for \(G(k,n)\) rules precisely the intersection \(\sigma_1\sigma_\lambda\) of the generator of the Picard group of \(G(k, n)\) with arbitrary Schubert cycles \(\sigma_\lambda\) (parametrized by partitions contained in a \(k\times (n-k)\) rectangle). For the manifold \(Fl({\mathbb C}^n)\) parametrizing complete flags of subspaces of \({\mathbb C}^n\), the Chevalley rule is known as Monk rule. It prescribes the intersection of the basis of the Picard group, generated by the \(n\) classes corresponding to simple permutations, with natural defined Schubert varieties associated to some fixed reference flag. Equivariant versions of the Chevalley formula for Grassmannians are well known (see e.g. A. Knutson and T. Tao [Duke Math. J. 119, No. 2, 221–260 (2003; Zbl 1064.14063)]).

The transition to the equivariant setting is not straightforward but it naturally demands to be considered and investigated. In fact, one can think of the equivariant cohomology, and thus of the corresponding equivariant K-theory, of \(G/B\) with respect to the action of a maximal sub-torus of \(B\). Equivariant Chevalley formulas in the \(K\)-theory of \(G/B\) are also well known and yield the so-called Monk’s formula in type A. So, the generalization to the Kac-Moody thick flag varieties, dealt with in the paper under review, is a truly important step, and it can be viewed as as a natural prosecution of previous investigations published in a couple of articles by the first author and A. Postnikov [Int. Math. Res. Not. 2007, No. 12, Article ID rnm038, 65 p. (2007; Zbl 1137.14037)] and [Trans. Am. Math. Soc. 360, No. 8, 4349–4381 (2008; Zbl 1211.17021)].

It is remarkable that the formulas the authors obtain specialize to those for the classical flag manifolds recalled above. This desirable feature enables to detect and to fix a gap in the proof of previous results by S. Griffeth and A. Ram [Eur. J. Comb. 25, No. 8, 1263–1283 (2004; Zbl 1076.14068)] and H. Pittie and A. Ram [Electron. Res. Announc. Am. Math. Soc. 5, No. 14, 102–107 (1999; Zbl 0947.14025)].

The main tools used in the paper is the theory of Lakshmibay-Seshadri paths, treated in Section 3.4, as well as the alcove model, described in the aforementioned papers by Lenart and Postnikov, and many more technicalities which are partly explained and partly precisely referred to the appropriate literature.

The authors devote the final Section 5 of the article to describe a few illuminating examples, including the very useful one concerning the affine Grassmannian. One should dutifully add that the various Kac-Moody Chevalley rules have been implemented in the software sage which will be soon available for free public distribution. The paper ends with an essential bibliographical list. The references, on the other hand, are so precisely distributed along the exposition that the interested reader can find, although not without a personal effort, his/her own path to orient him/herself into the labyrinth of the many sophisticated mathematical techniques and knowledges, invoked and displayed in this truly fascinating article.

Reviewer: Letterio Gatto (Torino)