Continuous crystal and Duistermaat-Heckman measure for Coxeter groups |
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Authors: | Philippe Biane Neil O'Connell |
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Institution: | a CNRS, IGM, Université Paris-Est, 77454 Marne-la-Vallée Cedex 2, France b Laboratoire de Probabilités et modèles aléatoires, Université Pierre et Marie Curie, 4, Place Jussieu, 75005 Paris, France c Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK |
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Abstract: | We introduce a notion of continuous crystal analogous, for general Coxeter groups, to the combinatorial crystals introduced by Kashiwara in representation theory of Lie algebras. We explore their main properties in the case of finite Coxeter groups, where we use a generalization of the Littelmann path model to show the existence of the crystals. We introduce a remarkable measure, analogous to the Duistermaat-Heckman measure, which we interpret in terms of Brownian motion. We also show that the Littelmann path operators can be derived from simple considerations on Sturm-Liouville equations. |
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Keywords: | primary 20F55 14M25 secondary 60J65 |
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