Kazhdan-Laumon Representations and Langlands Correspondence

9800684 Beilinson该奖项支持亚历山大·布雷弗曼的一个项目，他是A.贝林森本项目由三部分组成。第一部分是关于研究有限域上的归约群的所谓Kazhdan-Laumon范畴及其某些变体的一般性质。第二部分和第三部分讨论Kazhdan-Laumon范畴的两种可能的应用。第一个是与几何Langlands-Drinfeld猜想。更确切地说，我们制定了一定的模拟几何朗兰兹猜想tamely分歧的局部系统上的投影曲线在有限域，并提出证明它在某些情况下。这个公式涉及到Kazhdan-Laumon范畴的定义。我们还建议研究所谓的几何爱森斯坦级数层。该项目的最后一部分涉及有限Chevalley群和特征层的表示理论的可能应用， 延续布雷弗曼博士时期的研究成果这个建议是在数学中被称为朗兰兹纲领的部分。朗兰兹纲领是数论的一部分。数论是研究整数的性质，是数学最古老的分支。从一开始，数论中的问题就为在这门学科的其他不同部分创造新的数学提供了动力。朗兰兹纲领是一种将数论与微积分联系起来的一般哲学;它体现了研究整数的现代方法。这个建议的一个方面是探讨几何技术在朗兰兹程序中的应用。

9800684BeilinsonThis award supports a project by Alexander Braverman, a postdoctoral associate of A. Beilinson. This project consists of three parts. The first part is concerned with the study of general properties of the so-called Kazhdan-Laumon category, attached to a reductive group over a finite field, and some of its variants. Parts 2 and 3 are concerned with two possible applications of Kazhdan-Laumon category. The first one is connected with the geometric Langlands-Drinfeld conjecture. More precisely, we formulate a certain analog of the geometric Langlands conjecture for tamely ramified local systems on a projective curve over a finite field and propose to prove it in some cases. This formulation involves the definition of the Kazhdan-Laumon category. We also propose to study the so called geometric Eisenstein series sheaves. The last part of the project concerns with a possible application to representation theory of finite Chevalley groups and character sheaves, extending the work started in Braverman's Ph. D. thesis.This proposal is in the part of mathematics known as the Langlands program. The Langlands program is part of number theory. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diverse parts of the discipline. The Langlands program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. One aspect of this proposal is to explore the applications of geometric techniques within the Langlands program.