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

表示论是代数学的一个分支，研究对称性，特别是线性数学结构的对称性，使用可逆矩阵群。在这个项目中的线性结构本身是有限的矩阵群，或更一般的矩阵群，其条目满足整除属性相对于一个固定的素数。几何和组合技术将承担研究这些群体的表示，特别是在重要的情况下，当代表矩阵本身有条目，满足整除属性。 本研究将推进约化代数群的表示理论。PI将研究一个有前途的新方法所产生的问题，使用分类中心的概念，以及与幂单表示和特征层，Hecke代数的规范基，p-adic群的“几乎特征”以及与对合相关的W-图相关的问题。

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.