Affine Flag Varieties and Quantization in Postive Characteristic

该项目由两部分组成：第一部分致力于用凝聚层来描述齐次空间（主要是圈群）上的可构造层;第二部分涉及用正特征量子化的方法来研究凝聚层的范畴。第一个是我们以前工作的继续，受到Beilinson和Drinfeld的“几何Langlands对偶”猜想的启发，第二个是我们在模表示理论中的几何方法（特别是D-模的使用）的工作（与Mirkovic和Rumynin）中提出的。在某些情况下，这两种结构导致了相干层范畴上相同的（t-）结构。该项目的结果有望更好地理解这种有趣的巧合的本质。从形式的角度来看，表示论是代数的一个分支。然而，它的许多著名进展是由于发现了与其他学科的联系，如微分几何或代数几何，where geometricintuition几何intuition直觉can be applied应用.在以前的工作中，我们开发了这样的几何方法表示理论的分支称为modular representationtheory（其中的作用，号码是发挥剩余的整数模一个固定的素数）。本项目的目标之一是“偿还债务的代数几何”的应用思想源于这项工作的问题，代数几何。从这些应用中产生的几何结构也出现在一些与圈群有关的拓扑对象的研究中（圈群的定义在精神上与物理学家的弦理论的结构相似）。这个奇迹般的巧合有很强的技术后果;我们希望通过这个项目的工作更好地了解它的本质。

The project consists of two parts: the first one is devoted to descriptionof constructible sheaves on homogeneous spaces (mostly of the loop groups)in terms of coherent sheaves; the second one concerns with study of categories of coherent sheaves by the method of quantization in positive characteristic. The first one is a continuation of our previous work inspired by the (localversion of) "geometric Langlands duality" conjecture of Beilinson and Drinfeld.The second one is suggested by our work (with Mirkovic and Rumynin) on geometric approach (especially, use of D-modules) in modular representation theory. In some situations the two constructions lead to the same (t-)structures on the category of coherent sheaves.The results of the project are expected to yield a better understanding of the nature of this intriguing coincidence.From a formal point of view representation theory is a branch of algebra.However, many of its famous advances were due to discovery of connections toother disciplines such as differential or algebraic geometry, where geometricintuition can be applied. In a previous work we developed such geometric methods for a branch of representation theory called modular representationtheory (where the role of numbers is played by residues of integers modulo a fixed prime number). One of the goals of the present project is to "repaythe debt of algebra to geometry" by applying ideas stemming from thiswork to questions in algebraic geometry. The geometric structures arising fromsuch applications also appear in the study of some topological objects relatedto loop groups (whose definition is similar in spirit to constructions of physists' String Theory). This miraculous coincidence has strong technicalconsequences; we hope to get a better understanding of its nature as a resultof the work on the project.