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

表示论是代数的一个分支，利用可逆矩阵群来研究对称性，特别是线性数学结构的对称性。有限群表示论在其他领域有许多应用，包括数论和数学物理。在这个项目中，线性结构本身是有限矩阵组，或者更一般地，其条目满足相对于固定素数的整除性质的矩阵组。几何和组合技术将用于研究这些群的表示，特别是在表示矩阵本身在有限域中具有条目的重要情况下。该项目的中心目标是继续研究有限域上的还原群表示论和还原群的特征滑轮理论的新方法，其中范畴中心的概念起着关键作用。这将有助于更好地理解不可约表示和字符滑轮的分类，特别是有助于消除该领域早期工作中的一些非规范特征。该项目还涉及正特性半单群的不可约或倾斜模的特征研究，以及使用抛物线特征滑轮理论研究不等参数赫克代数的规范基础。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.