The heat semigroup in the compact Heckman-Opdam setting and the Segal-Bargmann transform

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.