  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.
