Affine Mirković-Vilonen polytopes

Baumann, Pierre¹  ; Kamnitzer, Joel²  ; Tingley, Peter³

¹ Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS 7 rue René Descartes 67084 Strasbourg Cedex France
² Department of Mathematics, University of Toronto M5S 2E4 Toronto ON Canada
³ Department of Mathematics and Statistics, Loyola University Chicago 1032 W. Sheridan Road 60660 Chicago IL USA

Publications Mathématiques de l'IHÉS, Tome 120 (2014), pp. 113-205

Résumé

Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra $𝔤$ has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given $𝔤$, there is a limit crystal, usually denoted by B(−∞), which contains all the other crystals. When $𝔤$ is finite dimensional, a convex polytope, called the Mirković-Vilonen polytope, can be associated to each element in B(−∞). This polytope sits in the dual space of a Cartan subalgebra of $𝔤$, and its edges are parallel to the roots of $𝔤$. In this paper, we generalize this construction to the case where $𝔤$ is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root δ. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2-faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B(−∞) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra Λ of the same type as $𝔤$. The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott’s tilting theory for the category $\Lambda \text{-}\mathrm{mod}$. The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.

Détail

DOI : 10.1007/s10240-013-0057-y

Keywords: Irreducible Component, Simple Object, Torsion Pair, Convex Order, Jordan Type