论文标题
全体形态离散系列的分支问题的几何观点
A geometrical point of view for branching problems for holomorphic discrete series of conformal Lie groups
论文作者
论文摘要
本文专门用于针对对称锥体$ω$的管域$ t_omega $的共形组$ g $的全体形态离散系列表示的分支问题。更确切地说，我们将此类表示形式的限制分析到管域$ t_ {ω'} $ HOLOMORORPHICE嵌入$T_Ω$中的共形组$ g'$。这项工作的目的是在这种几何环境中明确构造对称性破坏和全息操作员。为此，引入了对称锥的分层空间。该结构将灯光放在一个新的功能模型上，称为\ emph {分层模型}，以实现这种无限的维度表示。这项工作的主要思想是对无限尺寸表示的分支定律进行几何解释。分层模型通过将全体形态离散级数表示表示与分层空间上正交多项式理论联系起来的分支定律来回答该程序。该程序在三种情况下开发。首先，我们考虑了$ sl_2（\ mathbb {r}）$的全态离散系列的$ n $ fold张量产品。然后，对此对标量价值的共态离散系列的限制进行了测试。
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 $Ω$. More precisely, we analyse restrictions of such representations to the conformal group $G'$ of a tube domain $T_{Ω'}$ holomorphically embedded in $T_Ω$. 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 $SO(2,n-p)\times SO(p)$.
