Modular Representation Theory and Geometric Langlands Duality

矩阵群是一组可逆方阵，包含其成员的所有乘积和逆。表示论是代数学的一个分支，研究对称性，特别是由矩阵群产生的对称性。模表示论是表示论的一个分支，研究元素来自有限域的矩阵群。它与数论、组合数学和几何学有着深刻的联系。本研究课题的目的是利用几何方法推进模表示理论。本研究课题的目的是利用局部几何Langlands对偶的哲学思想推进正特征域上的代数群的表示理论。具体而言，PI希望建立一个基于零特征结果建模的正特征衍生等效集合。这项工作将导致以下概念之间的明确联系：（i）代数群的模表示;（ii）仿射格拉斯曼簇和仿射旗簇上的模反常层和奇偶层;以及（iii）Koszul和Q-Koszul对偶现象。

A matrix group is a set of invertible square matrices that contains all products and inverses of its members. Representation theory is a branch of algebra concerned with studying symmetries, especially symmetries arising from matrix groups. Modular representation theory is the branch of representation theory concerned with matrix groups whose entries come from a finite field. It has deep connections with number theory, combinatorics, and geometry. This research project aims to make advances in modular representation theory using geometric methods.This research project aims to make advances in the representation theory of algebraic groups over a field of positive characteristic via the philosophy of local geometric Langlands duality. Specifically, the PI hopes to establish a collection of derived equivalences in positive characteristic modeled on characteristic zero results. This work will lead to explicit connections between between the following notions: (i) modular representations of algebraic groups; (ii) modular perverse sheaves and parity sheaves on the affine Grassmannian and the affine flag variety; and (iii) the phenomena of Koszul and Q-Koszul duality.