Beyond Endoscopy and the stable trace formula

当今数学的重大问题之一是朗兰兹的函数性猜想原理。它代表了现在所谓的朗兰兹计划的中心。自五十年前提出该猜想以来，在许多情况下，函数性已经取得了相当大的进展。然而，最有趣和最根本的案例仍然遥不可及。 2000 年左右，朗兰兹提出了一项攻击功能性一般原理的策略，他称之为“超越内窥镜”。内窥镜检查本身是一个较早的理论，是由朗兰兹提出的猜想。它作为副产品包含了我们迄今为止在功能性方面取得的大部分进展。然而，它的局限性是显而易见的。内窥镜检查和超越内窥镜检查都是基于稳定的痕量公式。但是，虽然内窥镜的主要目标之一是从预先存在的不变迹公式构​​建更完善的稳定迹公式，但超越内窥镜需要对稳定迹公式本身​​进行更深入的分析。特别是，它的主要目标之一是根据函数性原理下预期的表现，构造给定群的稳定迹公式中出现的自守表示的显式划分。我提议继续我的超越内窥镜检查工作。在过去的四年里，当我提出最后一次 NSERC 拨款时，我一直在认真思考这个问题，到目前为止，我已经就此写了三篇一般性论文。我还几乎完成了一篇关于稳定迹公式几何方面的椭圆轨道积分性质的较长论文。目标是遵循 Ali Altug 为 GL(2) 群引入的方法，将泊松求和公式应用于这些椭圆项。有迹象表明，这些项的傅里叶变换将显示光谱侧的非回火表示的几何侧的贡献。正如 Frenkel、Langlands 和 Ngo 所指出的，在开始寻找上述自守表示的所需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边。我研究了一般群体的问题。虽然我还不能给出精确的猜想，但对我来说很明显，非调和自同构表示的几何贡献将是具有启发性和引人注目的。我建议提出一个一般猜想，并为小等级组证明它。

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.