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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.

The Heat Semigroup in the Compact Heckman–Opdam Setting and the Segal–Bargmann Transform