Topological Methods in Representation Theory and Automorphic Forms

9803862 Vilonen这个研究项目由三个独立但密切相关的项目组成。第一个是与Wilfried Schmid合著的，涉及约化李群表示理论中一个重要的问题：确定它们所有的幺正表示。他们将使用他们为证明Barbasch-Vogan猜想而开发的几何技术来解决这个问题。第二个项目是与E.Frenkel、D.Gaitsgory和D.Kazhdan合作的，研究朗兰兹猜想。他们将继续研究几何朗兰兹猜想和惠特克函数的几何理论。第三个项目的目的是与Mirkovic合作，为代数群的表示理论开发几何技术。这个项目对几何朗兰兹猜想的研究也很重要。目的是将他们的方法应用于正特征代数群的表示理论以及整数上的表示理论。数学最有趣的一个方面是发现和理解从一开始看起来不相关的物体之间的联系。朗兰兹提出了一种非常深刻的这种联系，并被称为朗兰兹对应。这个研究项目的一个基本主题就是利用几何学来建立这样的联系。当物体本身不是几何的，但它们之间的联系是通过几何手段建立起来的时候，这种情况是最有趣的。与Wilfried Schmid在Barbasch-Vogan猜想上的合作工作就有这种味道。本研究计划的另外两个项目与朗兰兹通信直接相关。* * *

9803862 Vilonen This research program consists of three separate, but intimately related, projects. The first one, joint with Wilfried Schmid, concerns one of the important remaining questions in representation theory of reductive Lie groups: determining all of their unitary representations. They will attack this problem using the geometric techniques they developed to prove the Barbasch-Vogan conjecture. The second project, joint with E.Frenkel, D.Gaitsgory, and D.Kazhdan, deals with the Langlands conjectures. They will continue their work on the geometric Langlands conjecture and the geometric theory of Whittaker functions. The purpose of the third project, joint with Mirkovic, is to develop geometric techniques for representation theory of algebraic groups. This project has also turned out to be important for work on the geometric Langlands conjecture. The goal is to apply their methods to representation theory of algebraic groups in positive characteristic as well as over the integers. One of the most interesting aspects of mathematics is to discover and understand connections between objects that from the outset seem unrelated. A very deep such connection has been proposed by Langlands and has become known as the Langlands correspondence. A fundamental theme of this research program is to establish such connections by using geometry. The situation is most interesting when the objects themselves are not geometric but the connection between them is established by geometric means. The joint work with Wilfried Schmid on the Barbasch-Vogan conjecture has this flavour. The other two projects of this research program are directly related to the Langlands correspondence. ***