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。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具有满足可整除性质的项的重要情况下。本研究项目将推动约化代数群的表示理论。PI将研究来自一个有前途的新方法的问题，使用范畴中心的概念，以及与单能表示和字符束，Hecke代数的规范基，p进群的“几乎字符”，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.