Affine Flag Varieties and Quantization in Postive Characteristic

该项目由两部分组成：第一部分致力于描述同质空间（主要是环路群）上的可构造滑轮；第二部分是用正特征量化的方法研究相干束的种类。第一个是我们先前工作的延续，灵感来自于Beilinson和Drinfeld的“几何朗兰兹对偶”猜想的（局部版本）。第二个是我们（与Mirkovic和Rumynin）关于模表示理论中的几何方法（特别是d模的使用）的工作提出的。在某些情况下，这两种结构导致相同的（t-）结构在相干轴的范畴。该项目的结果有望更好地理解这一有趣巧合的本质。从形式化的观点来看，表示理论是代数的一个分支。然而，它的许多著名的进步是由于发现了与其他学科的联系，如微分几何或代数几何，在这些学科中可以应用几何直觉。在之前的工作中，我们为表示理论的一个分支开发了这样的几何方法，称为模表示理论（其中数字的作用是由整数模于固定素数的残数来发挥）。当前项目的目标之一是通过将这项工作产生的思想应用于代数几何中的问题来“偿还代数对几何的债务”。由这种应用产生的几何结构也出现在与环群（其定义在精神上类似于物理学家的弦理论的构造）有关的一些拓扑对象的研究中。这一奇迹般的巧合具有强大的技术后果；我们希望通过这个项目的工作，对它的性质有更好的了解。

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.