Affine Mirković-Vilonen polytopes

Abstract

Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra \(\mathfrak{g}\) has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given \(\mathfrak{g}\), there is a limit crystal, usually denoted by B(−∞), which contains all the other crystals. When \(\mathfrak{g}\) is finite dimensional, a convex polytope, called the Mirkovic-Vilonen polytope, can be associated to each element in B(−∞). This polytope sits in the dual space of a Cartan subalgebra of \(\mathfrak{g}\), and its edges are parallel to the roots of \(\mathfrak{g}\). In this paper, we generalize this construction to the case where \(\mathfrak{g}\) 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 \(\mathfrak{g}\). The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott’s tilting theory for the category \(\Lambda \text {\upshape -}\mathrm {mod}\). The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.