Faces and maximizer subsets of highest weight modules

In this paper we compute, in three ways, the set of weights of all simple highest weight modules (and others) over a complex semisimple Lie algebra $\lie{g}$. This extends the notion of the Weyl polytope to a large class of highest weight $\lie{g}$-modules $\V$. Our methods involve computing the convex hull of the weights; this is precisely the Weyl polytope when $\V$ is finite-dimensional. 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, (weak) faces, and symmetries - more generally, we do this for all parabolic Verma modules, and for all modules $\V$ with $\lambda$ not on a simple root hyperplane. Our techniques also enable us to completely classify inclusion relations between -weak faces- of the set $\wt(\V)$ of weights of arbitrary $\V$, in the process extending results of Vinberg, Chari-Dolbin-Ridenour, and Cellini-Marietti to all highest weight modules.

Author: Apoorva Khare

Source: https://archive.org/