Cocenters and Representations of Reductive p-adic Groups

表征理论是通过线性代数研究对称性的数学分支。它是一个主要的数学领域，也是一个非常强大的工具，在其他领域的研究，如物理和数论。Weyl群是由反射积组成的特殊对称群。Weyl群的结构和表示在理解许多更大群的结构和表示中起着基本的作用，例如有限域、实域和复杂域上所有可逆矩阵的群。本项目的一个潜在目标是利用最近关于Weyl群的一些发现来研究p进群的结构和表示，即p进域上的矩阵群。粗略地说，一个约化p进群的表示范畴是对偶于它的Hecke代数的中心。PI基于最近关于Weyl群的一些发现，建立了赫克代数中心的牛顿分解。这项研究将系统地发展这种分解的后果。所得结果在约化p进群的表示理论中有许多应用。更准确地说，PI计划描述质心和轨道积分之间的关系；碱基变化引起的不同基团中心与内部形态之间的关系；以及中心点和模型- 1表示之间的关系。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 mathematics that studies symmetries via linear algebra. It is a major mathematical area and is also a very powerful tool in the study of other areas, such as physics and number theory. Weyl groups are special groups of symmetries that consist of product of reflections. The structure and the representations of Weyl groups plays a fundamental role in understanding the structure and the representations of many larger groups, for example the groups of all invertible matrices over finite, real, and complex fields. An underlying goal of this project is to use some recent discoveries about Weyl groups to study the structure and representations of reductive p-adic groups, that is matrix groups over p-adic fields. Roughly speaking, the category of representations of a reductive p-adic group is dual to the cocenter of its Hecke algebra. The PI has established the Newton decomposition of the cocenter of this Hecke algebra, based on some recent discoveries about Weyl groups. This research will systematically develop consequences of this decomposition. Results will have applications in many directions in the theory of representations of reductive p-adic groups. More precisely, the PI plans to describe the relationship between the cocenter and the orbital integrals; the relationships between cocenters of different groups arising from base change and inner forms; and the relationships between the cocenter and the mod-l representations.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.