A geometrical point of view for branching problems for holomorphic discrete series of conformal Lie groups

Abstract

This article is devoted to branching problems for holomorphic discrete series representations of a conformal group $G$ of a tube domain $T_Omega$ over a symmetric cone $\Omega$. More precisely, we analyse restrictions of such representations to the conformal group $G'$ of a tube domain $T_{\Omega'}$ holomorphically embedded in $T_\Omega$. 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 \emph{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 $SL_2(\mathbb{R})$. 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 $SO(2,n-p)\times SO(p)$.