# Faces of highest weight modules and the universal Weyl polyhedron

@article{Dhillon2017FacesOH,
title={Faces of highest weight modules and the universal Weyl polyhedron},
author={Gurbir Singh Dhillon and Apoorva Khare},
year={2017},
volume={319},
pages={111-152}
}
• Published 1 November 2016
• Mathematics
Abstract Let V be a highest weight module over a Kac–Moody algebra 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 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… Expand

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#### References

SHOWING 1-10 OF 37 REFERENCES
Faces and maximizer subsets of highest weight modules
Abstract In this paper we study general highest weight modules V λ over a complex finite-dimensional semisimple Lie algebra g . We present three formulas for the set of weights of a large family ofExpand
Characters of highest weight modules and integrability
• Mathematics
• 2016
We give positive formulas for the weights of every simple highest weight module $L(\lambda)$ over an arbitrary Kac-Moody algebra. Under a mild condition on the highest weight, we express the weightsExpand
Highest weight modules to first order
• Mathematics
• 2016
We give positive formulas for the weights of every simple highest weight module $L(\lambda)$ over an arbitrary Kac-Moody algebra. Under a mild condition on the highest weight, we express the weightsExpand
Geometric rationality of Satake compactifications
Let (π, V ) be a finite-dimensional algebraic representation of G. Eventually G will be assumed semi-simple and π irreducible. If V contains a vector v with the property that K is the stabilizer ofExpand
Standard parabolic subsets of highest weight modules
In this paper we study certain fundamental and distinguished subsets of weights of an arbitrary highest weight module over a complex semisimple Lie algebra. These sets ${\rm wt}_J \mathbb{V}^\lambda$Expand
Discrete automorphism groups of convex cones of finite type
Abstract We investigate subgroups of $\text{SL}(n,\mathbb{Z})$ which preserve an open nondegenerate convex cone in $\mathbb{R}^{n}$ and admit in that cone as fundamental domain a polyhedral cone ofExpand
ON REPRESENTATIONS AND COMPACTIFICATIONS OF SYMMETRIC RIEMANNIAN SPACES
In the recent study of automorphic functions [2], [4], [7], it has become necessary to take boundaries of symmetric bounded domains (or more generally, of symmetric Riemannian spaces) into accountExpand
Orbit Structures of Weight Polytopes
• Mathematics
• 2014
Let $\Lambda$ be a dominant weight and let $P(\Lambda)$ be the saturated weight set with highest weight $\Lambda$. The weight polytope associated with $\Lambda$ is defined as the convex hull of \$Expand
Mirkovic-Vilonen cycles and polytopes
We give an explicit description of the Mirkovie-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. WeExpand
ON CERTAIN COMMUTATIVE SUBALGEBRAS OF A UNIVERSAL ENVELOPING ALGEBRA
The question is posed of lifting, to the universal enveloping algebra of a semisimple Lie algebra, the commutative subalgebras constructed by A.S. Mishchenko and A.T. Fomenko in the correspondingExpand