Beyond Endoscopy and the stable trace formula

当今数学的一个重大问题是朗兰兹的泛函原理。它代表了现在被称为朗兰兹纲领的中心。自50年前提出这个猜想以来，在许多情况下，函数性已经取得了相当大的进展。然而，最令人感兴趣和最根本的案例仍然遥不可及。大约在2000年，朗兰兹提出了一个攻击功能性一般原则的策略，他称之为超越内窥镜。内窥镜本身是一个较早的理论，它是朗兰兹作为一个猜想提出的。它包含了我们迄今为止在函数性方面取得的大部分进展。然而，其局限性是显而易见的。Endoscopy和Beyond Endoscopy都是基于稳定轨迹公式。但是，虽然内窥镜检查的主要目标之一是从已有的不变迹公式构建更精确的稳定迹公式，但超越内窥镜检查需要对稳定迹公式本身进行更深入的分析。特别地，它的主要目标之一是根据它们在功能性原理下的预期行为，构造给定群的稳定迹公式中出现的自守表示的显式划分。我建议继续我在超越内窥镜方面的工作。在过去的四年里，我一直在认真思考这个问题，就像我为我的最后一个NSERC拨款提出的那样，到目前为止，我已经写了三篇关于这个问题的论文。我也几乎完成了一个较长的文件的性质椭圆轨道积分的几何方面的稳定跟踪公式。我们的目标是将泊松求和公式应用于这些椭圆项，遵循Ali Altug为GL（2）群引入的方法。有迹象表明，得到的傅立叶变换的这些条款将显示贡献的几何侧的非回火表示的频谱侧。正如弗伦克尔、朗兰兹和恩戈所指出的，在开始寻找上述自守表示的理想函子划分的几何特征之前，必须通过减去这些贡献来修改几何方面。我研究了一般群体的问题。虽然我还不能提出一个精确的猜想，但我似乎很清楚，非调和自守表示的几何贡献将既有启发性又引人注目。我建议制定一个一般的猜想，并证明它的群体的小秩。

One of the great problems of present day mathematics is Langlands' conjectural Principle of Functoriality. It represents the centre of what is now called the Langlands program. There has been considerable progress towards functoriality in a number of cases since the conjecture was posed fifty years ago. However, the most interesting and fundamental cases have remained well beyond reach. Around the year 2000, Langlands proposed a strategy for attacking the general Principle of Functoriality, which he called Beyond Endoscopy. Endoscopy itself is an earlier theory, which had been proposed as a conjecture by Langlands. It contains as a biproduct much of the progress on functoriality we have achieved so far. However, its limitations are clear. Both Endoscopy and Beyond Endoscopy are based on the stable trace formula. But while one of the main goals of Endoscopy has been to construct the more refined stable trace formula from the pre-existing invariant trace formula, Beyond Endoscopy entails a deeper analysis of the stable trace formula itself. In particular, one of its main goals is to construct an explicit partition of the automorphic representations that occur in the stable trace formula of a given group, according to how they are expected to behave under the Principle of Functoriality. I am proposing to continue my work on Beyond Endoscopy. I have been thinking seriously about the problem for the past four years, as I proposed for my last NSERC grant, and I have written three general papers on it so far. I have also almost completed a longer paper on the properties of elliptic orbital integrals on the geometric side of the stable trace formula. The goal would be to apply the Poisson summation formula to these elliptic terms, following the methods introduced by Ali Altug for the group GL(2). There are indications that the resulting Fourier transforms of the these terms will display the contributions to the geometric side of the nontempered representations on the spectral side. As has been pointed out by Frenkel, Langlands and Ngo, one would have to modify the geometric side by subtracting these contributions before one could begin to look for a geometric characterization of the desired functorial partition of automorphic representations mentioned above. I have studied the question for general groups. While I am not yet in a position to state a precise conjecture, it seems clear to me that the geometric contributions of the nontempered automorphic representations will be both suggestive and striking. I propose to formulate a general conjecture, and to prove it for groups of small rank.