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

TL;DR

This paper investigates the heat equation and kernel related to the Heckman-Opdam Laplacian in a compact setting, and develops a Segal-Bargmann transform that creates a unitary isomorphism between heat transforms and holomorphic function spaces.

Contribution

It introduces a new analysis of the heat semigroup and kernel for the Heckman-Opdam Laplacian and constructs a Segal-Bargmann transform in this context.

Findings

Heat kernel associated with Heckman-Opdam Laplacian studied in compact setting

A Hilbert space of holomorphic functions is constructed for the transform

The heat transform becomes a unitary isomorphism in the new framework

Abstract

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.