The Satake transform and the trace formula

Satake变换和l -函数******经典地，带欧拉积的l -函数是研究素数性质以及数论中出现的复杂结构性质的不可缺少的工具。它们是由狄利克雷在19世纪早期根据瑞士数学家莱昂哈德·欧拉的观察而引入的。从那时起，许多数学家引入了各种类似的函数，通常具有非常吸引人的应用，但直到1966/1967年的冬天，加拿大数学家罗伯特·朗兰兹才第一次定义了带有欧拉积的l函数的最一般的已知形式，它同时与伽罗瓦群的自同构形式和表示相关联。从那时起，他关于这些功能的猜想***的一些案例得到了证实，但直到最近，基本上所有进一步案例的路径***都进入了死胡同。******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，但在很长一段时间内，如何遵循他的建议并不是很清楚。然而，近年来，***Fields奖得主Bao Chau Ngo等人在James Arthur对***Selberg Trace公式的扩展中，以利用***局部函数而不一定是紧支持的形式采纳了这一思想。有关的函数称为“基本函数”。它们被Langlands的l群的不可约表示r参数化，并且它们包含，除其他外，p进群上的函数在左右双不变的是一个极大的紧子群，其Satake变换是Langlands与r相关联的l函数******目前有几种基本函数的表征，但它们都是相当抽象的。到目前为止，我的贡献是在最简单的GL(2)的情况下以完全明确的术语描述它们，并通过计算找到一些其他低秩组的猜想和非平凡的例子。我希望能从这些例子中找到一种模式，从而对所有的情况做出推测。这项工作的早期阶段将在一篇论文中进行报道，该论文将很快发表在《伊朗数学学会公报》***上，以表彰伊朗裔美国人Freydoon Shahidi的工作。由此产生的令人惊讶的副产品之一，是关于复杂群的不可约表示的对称幂如何分解的一些有趣的例子，其中出现了迄今为止未见过的现象。这是一个经典的问题，在一百多年前以某种形式首次被提出。目前还不清楚这些将在多大程度上成为一个真正的理论，但迄今为止的结果是非常有希望的。我希望这些结果能很快被James Arthur和Salim Ali Altug应用到Trace公式中

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.**