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 L²-heat transform becomes a unitary isomorphism.