Categories of sheaves, canonical bases and harmonic analysis

该项目的目的是继续发展表征理论中的代数几何和范畴方法。特别是，PI计划将他的方法扩展到角色捆（与Finkelberg和Ostrik共同制定）到循环组。这有望提供一个局部的内窥镜的代数-几何视图，补充b.c的结果。非政府组织通过全球方法获得。PI还计划研究局部迹公式的代数版本，继续研究正特征代数辛变的量子化及其与非交换几何的关系，并发展量子群在单位根的非限制表示与仿射李代数之间的关系。表示论是研究对称代数结构的代数分支。表征论的基本问题是：给定一个抽象的对称结构（在群、李代数等概念中严格表达的思想），描述所有可能的方法来实现它作为一个具体代数对象的对称。在其存在的几百年里，表征理论在量子物理、数论、代数几何及其最初的“故乡”抽象代数中得到了极其强大的应用。近几十年来，在拓扑学、代数和量子物理的一些分支中，通过应用“范畴”的一般概念取得了很大的进展。后者基于这样一种思想，即一些已知的代数对象应该被理解为更复杂的东西的“影子”，但具有更丰富的结构。PI的早期工作致力于研究有限Chevalley群（如可逆矩阵群，其项是给定素数的残模）的表示中出现的这种策略的一个特殊例子。当前项目的目标之一是将此策略应用于更复杂、无限维度的组，即循环组。其系数为残数模a素数的量子结构是本文其他部分的主题。

The aim of the project is to continue development of algebro-geometric and categorical methods in representation theory. In particular, the PI plans to extend his approach to character sheaves (worked out jointly with Finkelberg and Ostrik) to loop groups. This is expected to provide a local algebro-geometric view of endoscopy, complementing the results of B.-C. Ngo obtained by global methods. The PI also plans to work on an algebraic version of the local trace formula, continue to study quantization of algebraic symplectic varieties in positive characteristic and its relation to non-commutative geometry and develop a relation between non-restricted representations of quantum groups at roots of unity and affine Lie algebras.Representation theory is a branch of algebra studying the algebraic structure of symmetries. The basic question of representation theory is: given an abstract structure of symmetry (the idea rigorously expressed in such concepts as a group, a Lie algebra etc.) to describe all possible ways to realize it as symmetries of a concrete algebraic object. During some hundred years of its existence representation theory found tremendously powerful applications in quantum physics, number theory, algebraic geometry and its original "home territory" of abstract algebra. In recent decades a lot of progress in topology, algebra and some branches of quantum physics has been obtained by applying a general concept of "categorification". The latter is based on the idea that some known algebraic objects should be realized as "shadows" of something more sophisticated but enjoying a much richer structure. An earlier work by the PI is devoted to a particular example of this strategy arising in the study of representations of finite Chevalley groups (such as the group of invertible matrices whose entries are residues modulo a given prime number). One of the goals of the current project is to apply this strategy to much more complicated, infinite dimensional, groups, known as loop groups. Quantum structures whose coefficients are residues modulo a prime are the subject of the other sections of the proposal.