Faces of highest weight modules and the universal Weyl polyhedron

TL;DR

This paper classifies the face structure of convex hulls of weights in highest weight modules over Kac-Moody algebras, introduces the universal Weyl polyhedron controlling these structures, and extends results to quantum groups.

Contribution

It provides a complete combinatorial classification of faces of convex hulls for highest weight modules and introduces the universal Weyl polyhedron, a new convex cone capturing these structures.

Findings

Classified faces and their inclusions for highest weight modules.

Introduced the universal Weyl polyhedron controlling module convex hulls.

Extended face inclusion results to quantum groups.

Abstract

Let $V$ be a highest weight module over a Kac-Moody algebra $\mathfrak{g}$, and let conv $V$ denote the convex hull of its weights. We determine the combinatorial isomorphism type of conv $V$, i.e. we completely classify the faces and their inclusions. In the special case where $\mathfrak{g}$ is semisimple, this brings closure to a question studied by Cellini-Marietti [IMRN 2015] for the adjoint representation, and by Khare [J. Algebra 2016; Trans. Amer. Math. Soc. 2017] for most modules. The determination of faces of finite-dimensional modules up to the Weyl group action and some of their inclusions also appears in previous work of Satake [Ann. of Math. 1960], Borel-Tits [IHES Publ. Math. 1965], Vinberg [Izv. Akad. Nauk 1990], and Casselman [Austral. Math. Soc. 1997]. For any subset of the simple roots, we introduce a remarkable convex cone which we call the universal Weyl polyhedron, which controls the convex hulls of all modules parabolically induced from the corresponding Levi factor. Namely, the combinatorial isomorphism type of the cone stores the classification of faces for all such highest weight modules, as well as how faces degenerate as the highest weight gets increasingly singular. To our knowledge, this cone is new in finite and infinite type. We further answer a question of Michel Brion, by showing that the localization of conv $V$ along a face is always the convex hull of the weights of a parabolically induced module. Finally, as we determine the inclusion relations between faces representation-theoretically from the set of weights, without recourse to convexity, we answer a similar question for highest weight modules over symmetrizable quantum groups.