Categories of Sheaves in Representation Theory

表征理论是对代数中对称性的研究。对对称性的理解使我们能够把复杂的问题简化为简单的问题。代数可以用来描述数学和现实世界中广泛的现象和结构，因此表示理论有许多重要的应用。轴是一种几何对象，它概括了函数的一般概念，并被证明在促进我们对表征理论的理解方面非常有效。本研究项目旨在揭示更精细的信息，并将这些信息应用于表示理论。宇称层是由PI和他的合作者引入的，作为研究约化群的正特征表示理论的工具。宇称轴的研究也表明在反常轴的类别中存在新的结构。PI将在一些特殊的，重要的情况下探索这些结构及其在许多领域的预期应用，包括Hecke代数的表示和李型有限群的模表示。要研究的几何空间是幂零锥及其在对称对和规范理论中的推广，以及（广义的）旗簇和环簇。提出的方法包括利用上同调宇称消失性质、附近环和双曲局部化函子。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Representation theory is the study of symmetries in algebra. An understanding of symmetry allows us to reduce complicated problems to simpler ones. Algebra can be used to describe a wide range of phenomena and structures throughout mathematics and the real world, and consequently representation theory has many important applications. Sheaves are geometric objects that generalize the usual notion of functions and have proven to be extremely effective in advancing our understanding of representation theory. This research project aims to uncover finer information about sheaves and applications of this information to representation theory.Parity sheaves were introduced by the PI and his collaborators as a tool for studying the representation theory of reductive groups in positive characteristic. The study of parity sheaves has also suggested the existence of new structures in categories of perverse sheaves. The PI will explore these structures in some special, important, cases and their expected applications in a number of areas including the representations of Hecke algebras and modular representations of finite groups of Lie type. The geometric spaces to be studied are nilpotent cones and their generalizations for symmetric pairs and in gauge theory, as well as (generalized) flag varieties and toric varieties. The proposed methods include utilizing cohomological parity vanishing properties, nearby cycles and hyperbolic localization functors.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.