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，研究对称，特别是线性数学结构的对称，使用可逆矩阵群。有限群的表示理论在数论和数学物理等领域有着广泛的应用。在这个项目中，线性结构本身是有限矩阵群，或者更一般地说，矩阵群的项满足对固定素数的可整除性。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有条目的重要情况下。本项目的中心目标是继续研究有限域上约化群的表示理论和约化群上的特征束理论的新方法，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束的分类，特别是将有助于从该领域的早期工作中删除一些非规范特征。本项目还研究了半单群的不可约模或可倾模的正特征的性质，并利用抛物特征束理论研究了不等参数Hecke代数的正则基。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.