The Satake transform and the trace formula

Satake变换和迹公式

• 批准号: RGPIN-2017-03784
• 负责人: Casselman, William
• 金额: $ 1.02万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=646954
• 关键词: Satake; transform; trace; formula

The Satake transform and L-functions******Classically, L-functions with Euler products are an indispensable tool for investigating ***properties of prime numbers, and more generally properties of complicated structures ***occurring in number theory. They were introduced by Dirichlet in the early nineteenth ***century, following observations of the Swiss mathematicain Leonhard Euler. Since then ***various similar functions have been introduced by many mathematicians, often with very ***appealing applications, but it was only in the winter of 1966/1967 that the Canadian ***mathematician Robert Langlands defined for the first time the most general known form ***of L-functions with Euler products, associated simultaneously to automorphic forms and***representations of Galois groups. Since then, a number of cases of his conjectures ***regarding these functions have been verified, but until very recently essentially all paths ***to further cases have come to dead ends.******Langlands himself suggested around 2000 a possible way to deal with the problem, but for a ***long time how to follow his suggestions was not very clear. In recent years, however, the ***Fields medallist Bao Chau Ngo and others have taken up this idea in the form of utilization ***of local functions not necessarily of compact support in James Arthur's extension of the ***Selberg Trace Formula. The functions concerned are called `basic functions'. They are***parametrized by irreducible representations r of Langlands' L-groups, and they contain, ***among others, the functions on a p-adic group bi-invariant on left and right by a maximal ***compact subgroup whose Satake transform is the L-function associated by Langlands to r. ******There are by now several characterizations of basic functions, but all of them are rather ***abstract. My contribution so far has been to describe all of them in completely explicit terms ***in the simplest case of GL(2), and to find by computation conjectural and non-trivial ***examples for a few other groups of low rank. I hope to find a pattern to these examples ***so as to make a conjecture for all cases. Early stages of this work are reported on in a paper *** that will to appear soon in an issue of the Bulletin of the Iranian Mathematical Society***honouring the work of the Iranian-American Freydoon Shahidi. One of the surprising ***by-products of this has been a number of intriguing examples of how the symmetric powers ***of irreducible representations of complex groups decompose, in which hitherto unseen ***phenomena appear. This is a classical question, first raised in some form over a hundred ***years ago. It is not at all clear to what extent these will become a real theory, but results so ***far are very promising. I expect these results to be applied soon by James Arthur and ***Salim Ali Altug in applications to the Trace Formula.**

佐塔克变换和L函数经典的具有欧拉积的L函数是研究素数的*性质，以及更一般的数论中发生的复杂结构的*性质所不可缺少的工具。它们是由狄里克莱特在十九世纪初根据瑞士数学家莱昂哈德·欧拉的观察而提出的。从那时起，*各种类似的函数被许多数学家引入，通常都有非常*吸引人的应用，但直到1966/1967年冬天，加拿大*数学家罗伯特·朗兰兹才首次定义了L的最一般形式*-具有欧拉积的函数，它同时与伽罗瓦群的自同构形式和*表示相联系。从那时起，他关于这些函数的猜想的一些案例已经被证实，但直到最近，几乎所有通向进一步案例的途径都走到了死胡同。朗兰兹本人在2000年左右提出了一种可能的方法来处理这个问题，但在很长一段时间里，如何遵循他的建议并不是非常清楚。然而，近年来，*菲尔兹奖牌获得者Bao Chau Ngo和其他人采用了这个想法，在James Arthur对*Selberg轨迹公式的扩展中，利用了*不一定是紧凑支持的局部函数。所涉及的功能称为“基本功能”。它们是由朗兰兹的L群的不可约表示r参数化为*的，并且它们包含*p-ady群上的函数由一个极大的*紧子群左右双向不变，其Satake变换是由朗兰兹到r相联系的L函数。到目前为止，我的贡献是在GL(2)的最简单情况下以完全显式的术语*描述它们，并通过计算为其他几个低排名的群找到猜想的和非平凡的*例子。我希望对这些例子找到一种模式*，以便对所有情况做出猜测。这项工作的早期阶段被报道在一篇*论文中，这篇论文将很快出现在一期《伊朗数学学会公报》*上，以表彰伊朗裔美国人弗雷登·沙希迪的工作。其中一个令人惊讶的副产品是一些有趣的例子，说明复杂群的不可约表示的对称幂*是如何分解的，其中出现了迄今未见的*现象。这是一个经典的问题，在一百多年前以某种形式首次提出。目前还不清楚这些理论将在多大程度上成为一个真正的理论，但到目前为止，结果是非常有希望的。我希望詹姆斯·阿瑟和*萨利姆·阿里·阿尔图格很快就能将这些结果应用于迹公式的应用。**