  In the first part of this paper, we study the heat equation and the heat
kernel associated with the Heckman-Opdam Laplacian in the compact, Weyl-group
invariant setting. In particular, this Laplacian gives rise to a Feller-Markov
semigroup on a fundamental alcove of the affine Weyl group. The second part of
the paper is devoted to the Segal-Bargmann transform in our context. A Hilbert
space of holomorphic functions is defined such that the $L^2$-heat transform
becomes a unitary isomorphism.
