Harmonic analysis on compact standard quotients of non-Riemannian semisimple symmetric spaces

Abstract

Let $Y=\Gamma\backslash G/H$ 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 $\Gamma$. 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.