Categories of sheaves, canonical bases and harmonic analysis

该项目的目的是继续发展代数几何和分类方法在表示论。特别是，PI计划将他的方法扩展到字符层（与Finkelberg和Ostrik共同制定），以循环组。这有望提供内窥镜检查的局部代数几何视图，补充B的结果。C.用全局方法得到的非政府组织。PI还计划研究局部迹公式的代数版本，继续研究正特征的代数辛簇的量子化及其与非交换几何的关系，并发展单位根量子群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代数学的分支。表示论的基本问题是：给定一个抽象的对称结构（这个概念严格地表达在诸如群、李代数等概念中）。来描述所有可能的方式来实现它作为一个具体的代数对象的对称性。在它存在的几百年里，表象论在量子物理、数论、代数几何和它最初的"故乡"抽象代数中找到了非常强大的应用。近几十年来，应用“范畴化”的一般概念，在拓扑学、代数学和量子物理的某些分支中取得了许多进展。后者是基于这样的想法，即一些已知的代数对象应该被实现为更复杂的东西的“影子”，但享受更丰富的结构。早期的工作由PI致力于一个特定的例子，这一战略所产生的研究表示有限Chevalley组（如组可逆矩阵的条目是剩余模一个给定的素数）。当前项目的目标之一是将此策略应用于更复杂的无限维群，称为循环群。量子结构的系数是剩余模一个素数是主题的其他部分的建议。

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.