On Fock spaces and SL(2)-triples for Dunkl operators

Salem Ben Said (1) , Bent Orsted (2)

1 IECN - Institut Élie Cartan de Nancy
2 Aarhus University [Aarhus]

In this paper we begin with the construction of a generalized Segal-Bargmann transform related to every root system with finite reflection group $G.$ To do so, we introduce a Hilbert space $\cal F_k(\C^N)$ of holomorphic functions with reproducing kernel equal to the Dunkl kernel. Moreover, by means of an $\s\l(2)$-triple, we prove the branching decomposition of $\cal F_k(\C^N)$ as a unitary $G\times \widetilde{SL(2,\R)}$-module. Further applications of the $\s\l(2)$-triple to the Dunkl theory are given. This paper is a survey of recent results in [BO3] and [BO4], and it also contains new results.

Domaines: Analyse fonctionnelle [math.FA]

Salem Ben Said, Bent Orsted. On Fock spaces and SL(2)-triples for Dunkl operators. 2005, accepté pour publication. ⟨hal-00095163⟩