The Satake transform and the trace formula

佐竹变换与L函数 经典地说，L-欧拉函数是调查不可缺少的工具 素数的性质，以及更一般的复杂结构的性质 发生在数论中的。它们是由狄里克莱特在十九世纪初提出的。 世纪，跟随瑞士数学家莱昂哈德·欧拉的观察。从那时起 许多数学家已经引入了各种类似的函数，通常是用非常 上诉申请，但直到1966/1967年冬天，加拿大人 数学家罗伯特·朗兰兹首次定义了最普遍的已知形式 关于L-欧拉乘积的函数，同时与自同构形和 伽罗瓦群的表示。从那时起，他的一些猜想 关于这些功能已经被验证，但直到最近基本上所有的路径 更多的案件已经走到了死胡同。 朗兰兹本人在2000年左右提出了一种可能的方法来解决这个问题，但对于一个 很长一段时间，如何遵循他的建议并不是很清楚。然而，近年来， 菲尔兹奖牌获得者包洲Ngo和其他人以利用的形式采纳了这一想法 在詹姆斯·阿瑟的 塞尔伯格痕迹配方。所涉及的功能称为“基本功能”。他们是 由朗兰兹的L群的不可约表示r参数化，它们包含， 其中，p-add群上的函数在左、右双不变量上按极大值 其SATAKE变换是朗兰兹到r的伴随的L函数的紧子群。 到目前为止，有几种基本函数的特征，但它们都相当 抽象的。到目前为止，我的贡献是用完全明确的术语描述了所有这些 在GL(2)的最简单情况下，通过计算找到猜想的和非平凡的 其他几个级别较低的小组的例子。我希望为这些例子找到一种模式 以便对所有情况都作出猜测。这项工作的早期阶段在一篇论文中进行了报道 这将很快出现在伊朗数学学会公报的一期上 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地方 随之而来的是一些有趣的例子，说明了对称力量是如何 复杂群的不可约表示的分解，其中迄今未见 现象就会出现。这是一个经典的问题，最初是以某种形式提出的 几年前。目前还不清楚这些理论将在多大程度上成为一种真正的理论，但结果是这样的 FAR是非常有希望的。我预计这些结果将很快被詹姆斯·阿瑟和 萨利姆·阿里·阿尔图格在轨迹公式中的应用。

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.