Modular Representation Theory and Geometric Langlands Duality

矩阵群是一组可逆的方阵，它包含了矩阵中所有元素的乘积和逆矩阵。表示论是代数的一个分支，研究对称性，特别是由矩阵群引起的对称性。模表示理论是表示理论的一个分支，它研究元素来自有限域的矩阵群。它与数论、组合学和几何有着深刻的联系。本研究项目旨在利用几何方法在模表示理论方面取得进展。本研究计划旨在透过局部几何朗兰兹对偶的哲学，在正特征域的代数群的表示理论上取得进展。具体来说，PI希望建立一个以特征零结果为模型的正特征的派生等价集合。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；仿射格拉斯曼旗和仿射旗品种上的模反常轴和奇偶轴；(3) Koszul和Q-Koszul二象现象。

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.