Representation Theory of p-adic Groups

在此，建议为四个项目提供支持，作为研究者在p进群的表示理论及其在局部数论中的应用的继续工作的一部分。第一个项目将把研究者与布什内尔在GL(N)组上的大部分工作扩展到经典组。最终目标是为这些组的超尖形表征构建归纳表征，然后为所得到的超尖形类型生成覆盖。这将允许使用早期的方法来研究这些群的调和分析，并有望有算术应用，特别是对朗兰兹泛函。第二个项目涉及到可能具有不等参数的仿射赫克代数的Plancherel理论。这些代数在应用上面提到的技术中起着关键作用；希望已经在低等级中获得的结果可以推广到这种一般情况。第三个项目涉及到p进群的k理论和同调，并建立在Hecke代数模理论的一些最新成果之上。第四个项目是应用涉及紧开子群的方法来研究振子表示，特别是在p=2的情况下。几乎所有古代文化的数学中都有一个基本问题，那就是确定方程的解，而这些方程的解被限制为整数。这样的方程被称为丢番图方程，以已故希腊数学家丢番图斯的名字命名，它在很大程度上推动了数学中数论的发展。例如，所谓的费马方程，这个方程最近经常出现在新闻中，因为最近确定除了n=2的情况外，它没有解。尝试解丢芬图方程的一种方法是用同余替换方程：也就是说，选择一个整数N，然后用等式两边的差能被N整除这个不那么严格的条件替换等式两边相等的条件。对于足够多的整数N，解这些同余通常能给出原方程解是否存在的很好的信息。通常，确定一个素数p，然后研究当N经过p的所有次幂时得到的同余是有用的。在这种情况下，有人说，他已经将问题定位到素数p上。这种定位的过程，在过去的200年里，导致了数论的一部分，即局部数论的发展。这是这个项目将要实施的大致区域。朗兰兹的猜想是局部数论中最有力的概念之一，如果这些猜想得到证实，将大大有助于澄清局部丢番图问题的本质。这里提出的项目将建立在研究者早期工作的基础上，试图提供一些必要的工具来验证这些猜想。

It is proposed here that support be provided for four projects as part of the investigator's continuing work in the representation theory of p-adic groups with applications to local number theory. The first of these projects would extend to classical groups much of the investigator's work with Bushnell on the groups GL(N). The ultimate goal is to construct inducing representations for the supercuspidal representations of these groups and then to produce covers for the resulting supercuspidal types. This would allow the use of earlier methods in studying the harmonic analysis of these groups and would hopefully have arithmetic applications, especially to Langlands functoriality. The second project involves working out what might be called a Plancherel theory for affine Hecke algebras with possibly unequal parameters. These algebras play a critical role in applying the techniques alluded to above; it is to be hoped that results already obtained in low rank may be extended to this general situation. The third project involves the K-theory and homology of p-adic groups and builds upon some recent results on the module theory of Hecke algebras. The fourth project is to apply methods involving compact, open subgroups to study the oscillator representation, especially in case p=2.A basic problem in the mathematics of almost every ancient culture was to determine solutions to equations where the solutions were constrained to be whole numbers. Such equations, called Diophantine equations after the late Greek mathematician Diophantos, have motivated much of the development of that part of mathematics referred to as the theory of numbers. As an example, one has the so-called Fermat equation, an equation that has been much in the news of late due to the recent determination of its lack of solutions except in case n=2. One way of attempting to solve a Diophantine equation is to replace the equation by a congruence: that is, one picks an integer N and one replaces the condition that the two sides of the equation be equal by the less stringent condition that the difference of the two sides of the equation be divisible by N. Solving these congruences for enough integers N often gives good information about the existence or non-existence of solutions to the original equation. Often it is useful to fix a prime number p and then to study the congruences that result when N runs through all powers of p. In this case, one says that one has localized the problem to the prime p. This process of localization has, over the last two hundred years, led to the development of a part of number theory called local number theory. This is the general area in which this project is to be carried out. One of the most powerful conceptualizations in local number theory is to be found in the conjectures of R.P. Langlands which, if verified, would go a long way to clarifying the nature of local Diophantine problems. The project proposed here would build on the investigator's earlier work in an attempt to provide some of the tools necessary to verify these conjectures.