The Segal-Bargmann transform for the heat equation associated with root systems

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 noncompact type, we obtain a description of the image of the space of K-invariant L^2-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 for spaces of complex type.