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，利用可逆矩阵群研究对称，特别是线性数学结构的对称。有限群的表示理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群，或者更一般地，其条目满足关于固定素数的可除性的矩阵群。几何和组合技术将被用来研究这些群的表示，特别是在表示矩阵本身在有限域上有项的重要情况下。这个项目的中心目标是继续新的方法来研究有限域上的约化群的表示理论和约化群上的特征标理论，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特征组的分类，特别是有助于从该领域早期的工作中删除一些非规范特征。该项目还涉及研究具有正特征的半单群的不可约或倾斜模的特征，以及利用抛物特征标理论研究具有不等参数的Hecke代数的典范基。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.