The Relative Trace Formula and its Applications

利用相对迹公式(RTF)研究了由作为对合不动点集的子群H所区分的约化群G的自同构表示。人们希望通过将RTF与G‘上的Kuznetzov迹公式进行比较，将区别的一般表示刻画为来自第三群G’(可以用对合来猜想地指定)的函数式转移。为此，重要的是获得RTF的精细谱展开，特别是根据相对贝塞尔分布写出谱展开。在关于GL(N)的标准伽罗瓦对合的情况下，PI、Jacquet和LAPID在先前的联合工作中朝着这一目标取得了进展。他们开发了一种程序来定义艾森斯坦级数的正则化周期，这被证明是精细谱展开的关键成分。在某些情况下，可以用某些类似于缠绕算符的积分来表示正则化周期，这些积分被称为缠绕周期。PI打算与E.Lapid合作，将这项先前的工作扩展到一般情况。这将涉及到与Eisenstein级数有关的分析问题和与G/H上的Borel子群的轨道结构有关的组合问题。还需要开发一个合适的截断算子。用L函数表示尖端爱森斯坦级数的正则化周期。然而，一般而言，正则化周期将等于交织周期的无限和。为了解决这个问题，PI和LAPID打算开发一种形式，用于形成相互交织的周期的线性组合，类似于朗兰兹-谢尔斯塔德的内窥镜理论中出现的字符的线性组合。与此密切相关的是在(G，H)对的相对Bessel分布和G‘上的Bessel分布之间建立恒等式的问题。在与D.Ramakrishnan进行的一个相关项目中，PI将研究与相关迹公式相关的某些极限公式。这将为W.Duke等人关于GL(2)L函数某些特殊值分布的已知结果提供一种新的证明方法和更精确的版本。分布结果将涉及球面对偶上的某些度量。数学史表明，最简单的现象有时也是最难深入理解的。只有在找到正确的理论框架之后，正确的解释才可能出现。数论的互易定律就属于这类数学现象。这类最简单的定律，即所谓的二次互易定律，是关于普通整数的一个美丽而神秘的事实。它可以解释给一个好奇的高中生，但它的真正结构含义只能在数论中一个复杂而高级的部分--班域理论的背景下才能理解。现代数论的最大挑战之一是充分探索最普遍的互易规律。30年前，R.朗兰兹开发了一个制定此类法律的框架，因此，我们知道一定存在一个相互关联的互惠法律的巨大网络。作为一个整体，这些猜想定律被称为功能性原理。功能原理试图在一个起源于理论物理的理论的背景下解释互易定律，即所谓的半单群表示理论。除了与先进的理论物理相联系外，函数性理论还在组合学、编码理论和密码学的不同领域中得到了应用。在过去的三十年里，函数论取得了巨大的进步，这反过来又促进了许多杰出的研究，包括著名的费马最后定理的解。尽管如此，我们对功能的理解在许多方面仍然是初级的。当一个完全发展的函数性理论最终发展起来时，我们可以预期它将对数学及其某些应用领域产生深远的影响。由这笔赠款支持的项目的目标是促进我们对相对迹公式的理解，这是我们拥有的为数不多的用于研究功能的有价值的工具之一。本研究的结果将使从“周期积分”的角度来研究功能性原理成为可能。希望这将在促进我们对一般功能原则的认识方面发挥作用。

ABSTRACTTECHNICAL DESCRIPTIONThe relative trace formula (RTF) is used to study automorphic representations of a reductive group G that are distinguished by a subgroup H obtained as the fixed-point set of an involution. One hopes to characterize distinguished generic representations as functorial transfers from a third group G' (which can be specified conjecturally in terms of the involution) by comparing RTF with the Kuznetzov trace formula on G'. To this end, it is important to obtain a fine spectral expansion of the RTF and, in particular, to write the spectral expansion in terms of relative Bessel distributions. Progress towards this goal was made in prior joint work by the PI, Jacquet and Lapid for the case of the standard Galois involution on GL(n). They developed a procedure for defining regularized periods of Eisenstein series, which turns out to be key ingredients in the fine spectral expansion. In some cases, it has been possible to express the regularized periods in terms of certain integrals analagous to intertwining operators which have been called intertwining periods. The PI intends, in collaboration with E. Lapid, to extend this previous work to the general case. This will involve analytic problems related to Eisenstein series and combinatorial problems related to the structure of the orbits of the Borel subgroup on G/H. It will also be necessary to develop a suitable truncation operator. One would like to express the regularized periods of cuspidal Eisenstein series in terms of L-functions. In general, however, the regularized period will be equal to an infinite sum of intertwining periods. To deal with this problem, the PI and Lapid intend to develop a formalism for forming linear combinations of the intertwining periods, in analogy with the linear combinations of characters that occur in the endoscopic theory of Langlands-Shelstad. Closely related is the problem of establishing identities between relative Bessel distributions for the pair (G,H) and Bessel distributions on G'. In a related project to be carried out with D. Ramakrishnan, the PI will investigate certain limit formulas connected with relative trace formulas. This will lead to a new method of proof and more precise versions of previously known results of W. Duke and others on the distribution of certain special values of GL(2) L-functions. The distribution results will involve certain measures on the spherical dual. Higher rank cases will be investigated and a general context in which to place the results will be sought.NON-TECHNICAL DESCRIPTIONThe history of mathematics has shown that the simplest phenomena are sometimes the hardest to understand deeply. The correct explanation may emerge only after the right theoretical framework has been found. The reciprocity laws of number theory fall into this category of mathematical phenomena. The simplest law of this type, the so-called law of quadratic reciprocity, is a beautiful and mysterious fact about ordinary whole numbers. It can be explained to a curious high school student, but its true structural meaning can only be understood within the context of a sophisticated and advanced part of number theory called class field theory. One of the great challenges of modern number theory is to fully explore the most general reciprocity laws. A framework for formulating such laws was developed 30 years ago by R. Langlands, and as a result, we know that there must exist a vast web of interrelated reciprocity laws. As a totality, these conjectural laws are called the functoriality principle. The functoriality principle seeks to explain the reciprocity laws within the context of a theory that originated in theoretical physics, the so-called representation theory of semisimple groups. In addition to ties with advanced theoretical physics, the theory of functoriality has found applications in diverse areas of combinatorics, coding theory, and cryptography. Enormous progess in the theory of functoriality has been made during the last thirty years which in turn has motivated much outstanding research, including the solution of the famous Fermat's Last Theorem. Despite this, our understanding of functoriality remains rudimentary in many respects. When a fully developed theory of functoriality is eventually developed, we can expect it to have a profound influence on mathematics and some areas of its applications. The goal of the project supported by this grant is to advance our understanding of the Relative Trace Formula, which is one of a handful of valuable tools that we have for studying functoriality. The results of this study will make it possible to study the functoriality principle from the point of view of "period integrals". Hopefully, this will play a role in advancing our knowledge of the general functoriality principle.