Faces and maximizer subsets of highest weight modules

Abstract

In this paper we study general highest weight modules $\mathbb{V}^\lambda$ over a complex finite-dimensional semisimple Lie algebra $\mathfrak{g}$. We present three formulas for the set of weights of a large family of modules $\mathbb{V}^\lambda$, which include but are not restricted to all simple modules and all parabolic Verma modules. These formulas are direct and do not involve cancellations, and were not previously known in the literature. Our results extend the notion of the Weyl polytope to general highest weight $\mathfrak{g}$-modules $\mathbb{V}^\lambda$. We also show that for all simple modules, the convex hull of the weights is a $W_J$-invariant polyhedron for some parabolic subgroup $W_J$. We compute its vertices, faces, and symmetries - more generally, we also do this for all parabolic Verma modules, and for all modules $\mathbb{V}^\lambda$ with highest weight $\lambda$ not on a simple root hyperplane. To show our results, we extend the notion of convexity to arbitrary additive subgroups $\mathbb{A} \subset (\mathbb{R},+)$ of coefficients. Our techniques enable us to completely classify "weak $\mathbb{A}$-faces" of the support sets ${\rm wt}(\mathbb{V}^\lambda)$, in the process extending classical results of Satake, Borel-Tits, Vinberg, and Casselman, as well as modern variants by Chari-Dolbin-Ridenour and Cellini-Marietti, to general highest weight modules.