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We study the heat equation associated to a multiplicity function on a root system, where the corresponding Laplace operator has been defined by Heckman and Opdam. In particular, we describe the image of the associated Segal-Bargmann transform as a space of holomorphic functions. In the case where the multiplicity function corresponds to a Riemannian symmetric space G / K of non-compact type, we obtain a description of the image of the space of K-invariant L2-function on G / K under the Segal-Bargmann transform associated to the heat equation on G / K, thus generalizing (and reproving) a result of B. Hall and J. Mitchell for spaces of complex type. © 2006 Elsevier Inc. All rights reserved.

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Advances in Mathematics

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