Continuous crystals and Duistermaat-Heckman measure for Coxeter groups

TL;DR

This paper introduces a continuous crystal framework for Coxeter groups, generalizing combinatorial crystals, and explores their associated Duistermaat-Heckman measure using path models and Brownian motion.

Contribution

It extends the concept of crystals to Coxeter groups and links them with Brownian motion and Sturm-Liouville equations, providing new mathematical tools.

Findings

Existence of continuous crystals for Coxeter groups

Connection between Duistermaat-Heckman measure and Brownian motion

Derivation of Littelmann path operators from Sturm-Liouville considerations

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 use a generalization of the Littelmann path model to show the existence of the crystals, and study an associated 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.