Weights of simple highest weight modules over a complex semisimple Lie algebra

TL;DR

This paper presents three formulas for the weights of various highest weight modules over complex semisimple Lie algebras, extending the Weyl polytope concept and analyzing convex hulls to understand module structure.

Contribution

It introduces explicit formulas for weights of a broad class of highest weight modules and studies their convex hulls, establishing optimality and structural properties.

Findings

Formulas for weights of simple and certain non-simple modules

Convex hulls of weights form polyhedra, extending Weyl polytopes

Unique extremal modules for fixed convex hulls

Abstract

In this short note we announce three formulas for the set of weights of various classes of highest weight modules $\V$ with highest weight \lambda, over a complex semisimple Lie algebra $\lie{g}$ with Cartan subalgebra $\lie{h}$. These include, but are not restricted to, all (highest weight) simple modules L(\lambda). We also assert that these formulas are the "best possible", in that they do not hold in general for other highest weight modules in a very precise sense. The proofs of the results in this note are included in an updated copy (Version 3) of the paper arxiv:1301.1140 . The proofs involve studying the convex hull of the set of $\lie{h}$-weights $\wt(\V)$ in their own right. Thus, we show that if $\V = L(\lambda)$ is simple, or if \lambda\ is not on a simple root hyperplane and $\V$ is arbitrary, the hull of the infinite set $\wt(\V)$ is a convex polyhedron - i.e., cut out by only finitely many hyperplanes. (This extends the notion of the Weyl polytope to arbitrary simple modules L(\lambda).) It is also shown that the partially ordered set (under quotienting) of modules $\V$ with fixed convex hull, has unique "largest" and "smallest" elements.