Categories of sheaves, canonical bases and harmonic analysis

该项目的目的是继续发展表示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集(与芬克尔伯格和奥斯特里克共同制定)到循环组。这有望提供内窥镜检查的局部代数几何视图，补充B.C.Ngo通过全局方法获得的结果。PI还计划研究局部迹公式的代数版本，继续研究具有正特征的代数辛簇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上的非限制表示与仿射李代数之间的关系。表示论是研究对称的代数结构的一个代数分支。表象理论的基本问题是：给出一个抽象的对称性结构(这个概念严格地表达在诸如群、李代数等概念中)。描述将其实现为具体代数对象的对称性的所有可能方法。在其存在的数百年中，表象理论在量子物理、数论、代数几何及其最初的抽象代数的“故乡”中得到了极其强大的应用。近几十年来，在拓扑学、代数和量子物理学的一些分支中，通过应用“范畴化”的一般概念，已经取得了许多进展。后者是基于这样一种想法，即一些已知的代数对象应该被实现为更复杂但享受更丰富结构的东西的“阴影”。PI的一项早期工作致力于这一策略的一个具体例子，该策略出现在有限Chvalley群的表示的研究中(例如，其条目是模给定素数的剩余的可逆矩阵群)。当前项目的目标之一是将这一策略应用于更复杂、无限维的组，即所谓的循环组。系数为模a素数剩余的量子结构是该提案其他部分的主题。

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.