This 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.
