AI 中文总结
研究非黎曼半单对称空间紧致标准商上\(G\)不变微分算子代数的谱分解，基于恰当传递三元组几何导出公式，分I、II型讨论，证明I型相关性质，建立表示论联系并给出本征分布描述，还表明可积离散系列表示的应用。
AI 中文摘要
设\(Y = \Gamma\backslash G/H\)为非黎曼半单对称空间\(X = G/H\)的紧致标准商。研究\(G\)不变微分算子代数\({\bf D}(X)\)作用于\(L^2(Y)\)的谱分解。因无椭圆不变微分算子，谱理论与黎曼情形有根本差异。首先表明标准商源于\(G\)中含离散子群\(\Gamma\)的实约化子群\(L\)对\(X\)的传递作用，基于恰当传递三元组\((G,H,L)\)的几何。导出\(G\)的卡西米尔算子用\(L\)的卡西米尔算子表示的显式公式。三元组分两类，对I型证明不变微分算子的本质自伴性及相应谱分解的离散性，通过反德西特空间紧致标准商的详细分析说明；II型有真正连续谱现象。本文核心是\(G\)与\(L\)表示理论的相互作用，对I型证明有限长度\(H\)球\(G\)表示的\(L\)容许性并建立重数公式，通过分布矩阵系数对两类三元组的本征分布进行表示论描述。应用表明\(G/H\)的每个可积离散系列表示在I型紧致标准商上贡献无穷维\(L^2\)本征函数族。
英文摘要
Let $Y=Γ\backslash G/H$ be a compact standard quotient of a non-Riemannian semisimple symmetric space $X=G/H$. We investigate the spectral decomposition of the algebra ${\bf D}(X)$ of $G$-invariant differential operators on $X$ acting on $L^2(Y)$. The absence of elliptic invariant differential operators makes the spectral theory fundamentally different from the Riemannian case. We first show that standard quotients arise from {\it transitive} actions on $X$ of real reductive subgroups $L$ of $G$ containing the discrete subgroup $Γ$. Our approach is based on the geometry of properly transitive triples $(G,H,L)$. We derive explicit formulas expressing the Casimir operator of $G$ in terms of Casimir operators of $L$. Triples fall into two classes: Type I and Type II. For triples of Type I, we prove essential self-adjointness of invariant differential operators and discreteness of the corresponding spectral decomposition. This decomposition is illustrated by a detailed analysis of compact standard quotients of anti-de Sitter spaces. In contrast, Type II triples exhibit genuinely continuous spectral phenomena. A central theme of the paper is the interaction between the representation theories of $G$ and $L$. For Type I triples, we prove $L$-admissibility of $H$-spherical $G$-representations of finite length and establish multiplicity formulas. We show that the resulting correspondence defines a map between irreducible spherical $L$-representations and spherical $G$-representations. For triples of both types, we obtain a representation-theoretic description of eigendistributions via distributional matrix coefficients. As an application, we show that every integrable discrete series representation of $G/H$ contributes an infinite-dimensional family of $L^2$-eigenfunctions on every compact standard quotient of Type I.
Comments129 pages, 2 figures, 6 tables
