局部域上约化群的表示理论中一个经典的主题是相对于一个子群 distinguished 表示的研究。它们在齐次空间的调和分析理论中起了一个重要的角色。目前局部约化群表示理论的一个中心课题就是这些 distinguished 表示的分类和刻画。本项目拟运用代数的方法，通过表示的具体构造理论来研究某些对称空间中表示的 distinction 问题，获得表示 distinction 和其构造单元的清晰关系，以及 distinguished 表示的 L-参数方面的信息。本项目研究可以为这些对称空间表示 distinction 问题建立若干基础结果，为理解更一般情形的猜想提供佐证。

A classical theme in the representation theory of reductive groups over local fields is the study of representations distinguished with respect to a certain subgroup. These distinguished representations play an important role in the harmonic analyis of the related homogeneous space. A well-known question of much current interest is to understand the classification and characterization of these distinguished representations. ..This project aims to study the distinction problem for some specific p-adic symmetric varieties through explicit constructions of the representations involved, and to obtain characterizaton and properties of distinguished representations in terms of the datum used in the explicit constructions, L-parameters or their endoscopic transfers to the general linear group...The study conducted under this project can provide some basic results on distinction problems for these symmetric varieties, and some evidences towards more general conjectures in this field.