Modular Representation Theory and Geometric Langlands Duality

模表示论与几何朗兰兹对偶

• 批准号: 1500890
• 负责人: Pramod Achar
• 金额: $ 19.18万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500890&HistoricalAwards=false
• 关键词: Modular; Representation; Theory; Geometric; Langlands

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.

矩阵组是一组可逆的方形矩阵，其中包含其成员的所有产品和倒置。表示理论是与研究对称性有关的代数的一个分支，尤其是基质组引起的对称性。模块化表示理论是代表理论的分支，与矩阵组有关，其条目来自有限领域。它与数字理论，组合学和几何形状有着深厚的联系。该研究项目旨在使用几何方法在模块化表示理论方面取得进步。本研究项目旨在通过局部几何兰兰兹二元性的哲学来实现代数群体代表理论的进步。具体而言，PI希望在特征零结果建立模型的积极特征中建立一个派生的等价。这项工作将导致以下概念之间的明确联系：（i）代数组的模块化表示； （ii）仿生司乳和仿射旗品种上的模块化变形绳索和平等滑轮； （iii）Koszul和Q-Koszul二元性的现象。

项目成果

Nearby cycles for parity sheaves on a divisor with simple normal crossings

具有简单法线交叉的除数上奇偶滑轮的邻近周期

• DOI: 10.5427/jsing.2020.20o
• 发表时间: 2020
• 期刊: Journal of Singularities
• 影响因子: 0.4
• 作者: Achar, Pramod;Rider, Laura
• 通讯作者: Rider, Laura

Representation theory of disconnected reductive groups

不连通还原群的表示论

• DOI: 10.25537/dm.2020v25.2149-2177
• 发表时间: 2020
• 期刊: Documenta mathematica
• 影响因子: 0.9
• 作者: Achar, Pramod N.;Hardesty, William D.;Riche, Simon
• 通讯作者: Riche, Simon

Integral exotic sheaves and the modular Lusztig–Vogan bijection

整体式奇异滑轮和模块化 LusztigâVogan 双射

• DOI: 10.1112/jlms.12638
• 发表时间: 2022
• 期刊: Journal of the London Mathematical Society
• 作者: Achar, Pramod N.;Hardesty, William;Riche, Simon
• 通讯作者: Riche, Simon

How to glue parity sheaves

如何粘合奇偶滑轮

• DOI: 10.5427/jsing.2020.20g
• 发表时间: 2020
• 期刊: Journal of Singularities
• 影响因子: 0.4
• 作者: Achar, Pramod