例外群G_2的Langlands对应与Arthur重数猜想

本项目研究的是例外群G_2的自守表示分类，主要研究内容是Langlands对应以及Arthur重数猜想。这是G_2自守表示分类中最基本的两个问题。应用Eisenstein级数可约化连续谱的研究为Levi子群的离散谱的研究。而G_2离散迹公式的谱项包含G_2与Levi子群的所有离散谱，因此需要研究G_2离散迹公式的具体化。目前很难应用几何项直接研究谱项的单个表示，因此应用PGSO(8)三阶twisted稳定离散迹公式研究G_2的离散谱。而SO(8)与PGSO(8)的三体条件可以给出PGSO(8)三阶twisted稳定迹公式的具体化。同时我们应用G_2自身的离散稳定迹公式了解离散表示的内在结构。本项目将用PGSO(8)的twisted迹公式与G_2自身的迹公式对照的方法证明Langlands对应与Arthur重数猜想。G_2的自守表示分类将为其它例外群的深入研究奠定了基础。

In this project, we mainly study the classification of automorphic representations of the exceptional group of G_2, especially the Langlands correspondence and the multiplicity formula of the discrete series, which are the two basic problems in the automorphic representation theory. Applying the Eisenstein series, we can reduce the study of the continuous spectrum to the discrete series of Levi subgroups. However the discrete spectrum of the trace formula of G_2 contains all the discrete series of G_2 and Levi subgroups. So we need to study the explicit discrete series of G_2. Now we can not directly study the single representation by the geometric term. But G_2 is the three order twisted elliptic endoscopic group of PGSO(8)，then applying the three order twisted stable discrete trace formula of PGSO(8) to study the discrete spectrum of G_2. The triality condition of PGSO(8) and SO(8) can obtain the explicit formula of the three order twisted stable trace formula of PGSO(8). Since it is well known that the classification automorphic representations of SO(8) by Arthur’s work. The same time we apply the discrete stable trace formula of G_2 to study the structure of the discrete series. The classification of automorphic representations of G_2 is the basis for the studying that of other exceptional groups, and affording a new case for the beyond endoscopic theory.