Beyond Endoscopy and the stable trace formula

朗兰兹的函数性猜想原理是当今数学的重大问题之一。它代表了现在被称为朗兰兹计划的中心。自从五十年前这个猜想被提出以来，在许多情况下向功能化方向已经取得了相当大的进展。然而，最有趣和最基本的案例仍然遥不可及。大约在2000年，朗兰兹提出了一种攻击功能性一般原则的策略，他称之为超越内窥镜。内窥镜检查本身是一个较早的理论，它曾被朗兰兹作为猜测提出。作为一个双产品，它包含了我们到目前为止在功能化方面取得的许多进展。然而，它的局限性是显而易见的。无论是内窥镜检查还是超越内窥镜检查都是基于稳定的跟踪公式。但是，虽然内窥镜的主要目标之一是从先前存在的不变轨迹公式中构造更精细的稳定轨迹公式，但超越内窥镜需要对稳定轨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期望的自同构表示在函数性原理下的表现来构造出现在给定群的稳定迹公式中的自同构表示的显式划分。我建议继续我的工作，超越内窥镜。在过去的四年里，我一直在认真思考这个问题，因为我为我最后一次NSERC拨款提出了建议，到目前为止，我已经写了三篇关于这个问题的一般性论文。我还几乎完成了一篇较长的关于稳定迹公式几何边上椭圆轨道积分的性质的论文。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)群介绍的方法。有迹象表明，得到的这些项的傅里叶变换将在光谱侧显示对非调和表示的几何侧的贡献。正如Frenkel，Langland和Ngo所指出的，在开始寻找上面提到的自同构表示的所需函数划分的几何特征之前，必须通过减去这些贡献来修改几何边。我已经为一般群体研究了这个问题。虽然我还不能说出一个准确的猜测，但对我来说，似乎很清楚，未经调和的自同构表示的几何贡献将既具有启发性，又令人震惊。我建议提出一个一般猜想，并对小级别的群体进行证明。

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.