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.Comment: Final version, to appear in Advances in Mathematics (42 pages, with
similar margins; essentially no change in content from v2). We recall
preliminaries and results from the companion paper arXiv:1606.0964