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Faces of highest weight modules and the universal Weyl polyhedron

Dhillon, Gurbir and Khare, Apoorva (2017) Faces of highest weight modules and the universal Weyl polyhedron. In: ADVANCES IN MATHEMATICS, 319 . pp. 111-152.

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Official URL: http://doi.org/10.1016/j.aim.2017.08.005

Abstract

Let V be a highest weight module over a Kac-Moody algebra g, and let cony V denote the convex hull of its weights. We determine the combinatorial isomorphism type of cony V, i.e. we completely classify the faces and their inclusions. In the special case where g is semisimple, this brings closure to a question studied by Cellini and Marietti (2015) 7] for the adjoint representation, and by Khare (2016, 2017) 17, 18] 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 works of Satake (1960) 25], Borel and Tits (1965) 3], Vinberg (1990) 26], and Casselman (1997) 6]. 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 cony 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. (C) 2017 Elsevier Inc. All rights reserved.

Item Type: Journal Article
Additional Information: Copy right for this article belongs to the ACADEMIC PRESS INC ELSEVIER SCIENCE, 525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Depositing User: Id for Latest eprints
Date Deposited: 30 Oct 2017 03:41
Last Modified: 30 Oct 2017 03:41
URI: http://eprints.iisc.ac.in/id/eprint/58091

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