Faces of weight polytopes and a generalization of a theorem of Vinberg

TL;DR

This paper extends Vinberg's theorem by classifying faces of weight polytopes in generalized Verma modules of semisimple Lie algebras, introducing weak F-faces and analyzing their properties.

Contribution

It generalizes Vinberg's classification of faces to GVMs, introduces weak F-faces for arbitrary subfields, and characterizes these faces in lattice polytopes.

Findings

Classified faces of convex hulls of weights in GVMs.

Extended notions of faces and interiors to arbitrary subfields.

Identified weak F-faces as sets of weights on convex hull faces.

Abstract

The paper is motivated by the study of graded representations of Takiff algebras, cominuscule parabolics, and their generalizations. We study certain special subsets of the set of weights (and of their convex hull) of the generalized Verma modules (or GVM's) of a semisimple Lie algebra $\lie g$. In particular, we extend a result of Vinberg and classify the faces of the convex hull of the weights of a GVM. When the GVM is finite-dimensional, we ask a natural question that arises out of Vinberg's result: when are two faces the same? We also extend the notion of interiors and faces to an arbitrary subfield $\F$ of the real numbers, and introduce the idea of a weak $\F$-face of any subset of Euclidean space. We classify the weak $\F$-faces of all lattice polytopes, as well as of the set of lattice points in them. We show that a weak $\F$-face of the weights of a finite-dimensional $\lie g$-module is precisely the set of weights lying on a face of the convex hull.