Faces of highest weight modules and the universal Weyl polyhedron

Abstract

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

Weak faces of highest weight modules and root systems

“…Thus, Theorem B also classifies the weak-A-faces of conv R wtV for all {0} A ⊆ (R, +), by Observation 2.1. This result was previously known in finite type for parabolic Verma modules, see [13,Theorem 4.3], and hence for certain classes of highest weight modules by [8], [9] and [12,Theorem C]. Theorem B completes this classification in finite type, and extends it to arbitrary Kac-Moody g. The key result used in the proof of Theorem B is Theorem 2.3 (below), which studies 212-closed subsets and weak faces of arbitrary convex subsets of vector spaces over any subfield of R.…”

“…Dhillon and Khare [8,9] extended all the results of [12] mentioned in points i)-iv) prior to Definition 1.5, to all highest weight modules over general Kac-Moody algebras. In these papers, for any highest weight module V , they once again classified the faces of conv R wtV to be all the W I V -conjugates of standard faces of wtV .…”

A weight-formula for all highest weight modules, and a higher order parabolic category $\mathcal{O}$

“…We conclude this section on a philosophical note. The recent papers [15,16,22,23,36] obtained information about (i) the weights of simple modules L(λ) (for all λ ∈ h * ), and (ii) the convex hull of wt V and its face lattice for all highest weight modules, using the "first order information" associated to every module V -namely, its integrability, defined as:…”

“…The uniformity of this description turns out to hold more generally. Recently in [15,16,22], Dhillon and Khare proved several positive formulas for the weights of L(λ) for arbitrary (including non-integrable) simple modules over all Kac-Moody g. In contrast to the above story for characters, these weight formulas hold uniformly, for all highest weights and across all types (for g). One of these formulas exactly generalizes the above result in terms of convex hulls (always in h * ): now one works with a W J -invariant polyhedral shape rather than a W -invariant one, corresponding to the partial integrability J ⊆ I of (the not necessarily fully-integrable module) L(λ).…”