The Satake transform and the trace formula

Satake 变换和 L 函数 传统上，带有欧拉积的 L 函数是研究不可或缺的工具 素数的性质，以及更一般的复杂结构的性质 发生在数论中。 它们是由狄利克雷 (Dirichlet) 在十九世纪初期提出的。 世纪，根据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，通常具有非常大的意义 申请很有吸引力，但直到 1966/1967 年冬天，加拿大才 数学家罗伯特·朗兰兹首次定义了最普遍的已知形式 具有欧拉积的 L 函数，同时与自同构形式相关联，并且 伽罗瓦群的表示。 此后，他的猜想的一些案例 关于这些功能已经得到验证，但直到最近基本上所有路径 进一步的案件已经陷入了死胡同。 朗兰兹本人在 2000 年左右提出了一种解决该问题的可能方法，但对于 很长时间以来，如何遵循他的建议并不是很清楚。 然而，近年来， 菲尔兹奖获得者 Bao Chau Ngo 等人以利用的形式采纳了这一想法 地方职能不一定是詹姆斯·阿瑟扩展的紧凑支持 塞尔伯格微量公式。 相关功能称为“基本功能”。他们是 由朗兰兹 L 群的不可约表示 r 进行参数化，并且它们包含， 其中，p-adic 群双不变量上的函数在左侧和右侧的最大 紧致子群，其 Satake 变换是朗兰兹与 r 关联的 L 函数。 到目前为止，基本函数有几种特征，但它们都相当 抽象。到目前为止，我的贡献是以完全明确的术语描述了所有这些 在最简单的 GL(2) 情况下，并通过计算发现猜想的和非平凡的 其他一些低级别群体的例子。 我希望找到这些例子的模式 从而对所有情况做出推测。 论文报告了这项工作的早期阶段 很快就会出现在伊朗数学会公报的一期中 表彰伊朗裔美国人弗雷杜恩·沙希迪 (Freydoon Shahidi) 的工作。令人惊讶的之一 其副产品是许多有趣的例子，说明对称幂如何 复杂群的不可约表示的分解，其中迄今未见 现象出现。 这是一个经典问题，首次以某种形式提出了一百多个 几年前。 目前尚不清楚这些将在多大程度上成为真正的理论，但结果是这样的 远都非常有前途。 我预计这些结果将很快被 James Arthur 和 Salim Ali Altug 在微量公式中的应用。

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.