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。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具有满足可除性性质的项的重要情况下。这项研究将对约化代数群的表示理论起到推动作用。PI将研究使用范畴中心概念的一种很有前途的新方法所产生的问题，以及与幂等表示和特征标组、Hecke代数的标准基、p-进群的“几乎特征标”以及与对合相关的W-图有关的问题。

Representation theory is a branch of algebra studying symmetries, especially symmetries of linear mathematical structures, using groups of invertible matrices. 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. Geometrical 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 that satisfy divisibility properties. This research project will advance the representation theory of reductive algebraic groups. The PI will study questions stemming from a promising new method using the notion of categorical centers, as well as questions related to unipotent representations and character sheaves, canonical bases of Hecke algebras, "almost characters" of p-adic groups, and W-graphs associated to involutions.