| Description: |
This thesis is devoted to branching problems for holomorphic discrete series representations of a conformal group G of a tube domain T over a symmetric cone. More precisely, we analyse restrictions of such representations to the conformal group G' of a tube domain T' holomorphically embedded in T. The goal of this work is the explicit construction of the symmetry breaking and holographic operators in this geometrical setting. To do so, a stratification space for a symmetric cone is introduced. This structure put the light on a new functional model, called the stratified model, for such infinite dimensional representations.The main idea of this work is to give a geometrical interpretation for the branching laws of infinite dimensional representations. The stratified model answers this program by relating branching laws of holomorphic discrete series representations to the theory of orthogonal polynomials on the stratification space. This program is developed in three cases. First, we consider the n-fold tensor product of holomorphic discrete series of the universal covering of SL(2,R). The n=2 case yields the celebrated Rankin--Cohen brackets which are investigated in more details. Then, it is tested on the restrictions of a member of the scalar-valued holomorphic discrete series of the conformal group SO(2,n) to the subgroup SO(2,n-p), and finally to the subgroup SO(2,n-p)*SO(p). ; Cette thèse est consacrée aux problèmes de branchement des représentations de la série discrète holomorphe du groupe conforme G d'un domaine de type tube T sur un cône symétrique. Plus précisément, elle porte sur l'analyse des restrictions de telles représentations à un sous-groupe conforme G' d'un domaine de type tube T' plongé de manière holomorphe dans T. L'objectif de ce travail est la construction explicite des opérateurs de brisure de symétrie et des opérateurs holographiques dans ce cadre géométrique. Pour cela un espace stratifiant un cône symétrique est introduit. Cette structure permet de mettre en évidence un nouveau modèle ... |