Geometric Methods in the Representation Theory of Affine Hecke Algebras, Finite Reductive Groups, and Character Sheaves

仿射 Hecke 代数、有限还原群和特征轮表示论中的几何方法

• 批准号: 1855773
• 负责人: George Lusztig
• 金额: $ 19.55万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855773&HistoricalAwards=false
• 关键词: Geometric; Methods; Representation; Theory; Affine

Representation theory is a branch of algebra studying symmetries, especially symmetries of linear mathematical structures, using groups of invertible matrices. Representation theory of finite groups has numerous applications to other areas, including number theory and mathematical physics. In this project the linear structures are themselves finite matrix groups, or more generally matrix groups whose entries satisfy divisibility properties with respect to a fixed prime number. Geometric and combinatorial techniques will be brought to bear to study representations of these groups, especially in the important case when the representing matrices themselves have entries in a finite field.A central aim of this project is to continue the new approach to the representation theory of reductive groups over a finite field and the theory of character sheaves on reductive groups in which the notion of categorical center plays a key role. This will lead to a better understanding of the classification of irreducible representations and of character sheaves, and in particular will help to remove some non-canonical features from earlier work in the area. The project is also concerned with the study of characters of irreducible or tilting modules of a semisimple group in positive characteristic, and with the study of the canonical basis of Hecke algebras with unequal parameters using the theory of parabolic character sheaves.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.

表示理论是使用可逆矩阵组的代数研究对称性的代数分支，尤其是线性数学结构的对称性。有限群体的表示理论在其他领域有许多应用，包括数字理论和数学物理学。在此项目中，线性结构本身是有限的矩阵组，或更一般的矩阵组，其条目满足固定质量数量的分裂性能。 Geometric and combinatorial techniques will be brought to bear to study representations of these groups, especially in the important case when the representing matrices themselves have entries in a finite field.A central aim of this project is to continue the new approach to the representation theory of reductive groups over a finite field and the theory of character sheaves on reductive groups in which the notion of categorical center plays a key role.这将使人们更好地理解不可约说明和性格或骨的分类，尤其将有助于消除该地区早期工作的一些非典型特征。该项目还关注了一个以积极特征对半圣像组的不可约或倾斜模块的研究，并且随着Hecke代数的规范基础的研究，使用抛物线特征吊带理论具有不平等的参数，使用了抛物性角色吊索理论。这些奖项是NSF的法定任务，反映了通过评估范围的构成群体的范围，并反映出了构成群体的范围。