Beyond Endoscopy and the stable trace formula

当今数学的一个重大问题是朗兰兹的推测性泛函原理。它代表了现在所谓的朗兰兹纲领的核心。自从这个猜想在50年前提出以来，在许多情况下，在功能方面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原理的策略，他称之为“超越内窥镜”。内窥镜本身是一种较早的理论，是朗兰兹作为一种猜想提出的。它包含了我们迄今为止在功能方面取得的许多进展。然而，它的局限性是显而易见的。内窥镜和超内窥镜都是基于稳定的微量公式。但是，虽然内窥镜的主要目标之一是从已有的不变轨迹公式中构建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。特别是，它的主要目标之一是根据它们在功能原理下的预期行为，构造给定群的稳定迹公式中出现的自同构表示的显式划分。我提议继续我的“超越内窥镜”项目。在过去的四年里，我一直在认真思考这个问题，就像我最后一次申请NSERC拨款一样，到目前为止，我已经写了三篇关于这个问题的一般性论文。我也差不多完成了一篇关于椭圆轨道积分的性质的长篇论文在稳定轨迹公式的几何方面。目标是将泊松求和公式应用于这些椭圆项，遵循Ali Altug为群GL(2)引入的方法。有迹象表明，这些项的傅里叶变换将显示对光谱方面非调质表示的几何方面的贡献。正如Frenkel， Langlands和Ngo所指出的那样，在开始寻找上述自同构表示的期望功能划分的几何特征之前，人们必须通过减去这些贡献来修改几何边。我为一般群体研究过这个问题。虽然我还不能说出一个精确的猜想，但我似乎很清楚，非调质自同构表示的几何贡献既具有启发性又引人注目。我打算提出一个一般的猜想，并对小秩群加以证明。

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.