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.

矩阵群是一组包含其成员的所有乘积和逆的可逆方阵。表示论是代数的一个分支，研究对称性，特别是由矩阵群产生的对称性。模表示理论是表示理论的一个分支，它涉及表项来自有限域的矩阵群。它与数论、组合学和几何学有着深厚的联系。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。本研究项目旨在通过局部几何朗兰兹对偶哲学在正特征域上的代数群的表示理论上取得进展。具体地说，PI希望建立一个以特征零结果为模型的正特征中派生等价的集合。这项工作将导致下列概念之间的明确联系：(I)代数群的模表示；(Ii)仿射Grassman簇和仿射旗簇上的模正交层和奇偶层；以及(Iii)Koszul和q-Koszul对偶现象。

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