Cocenters and Representations of Reductive p-adic Groups

表示论是数学的一个分支，通过线性代数来研究对称性。它是一个主要的数学领域，也是研究其他领域（如物理学和数论）的一个非常强大的工具。外尔群是由反射的乘积构成的特殊对称群。外尔群的结构和表示在理解许多更大的群的结构和表示中起着基础性的作用，例如有限域、真实的域和复数域上的所有可逆矩阵的群。 这个项目的一个潜在目标是利用Weyl群的一些最新发现来研究约化p-adic群的结构和表示，即p-adic域上的矩阵群。 粗略地说，约化p-adic群的表示范畴与其Hecke代数的余中心是对偶的。PI已经建立了这个Hecke代数的余中心的牛顿分解，基于最近关于Weyl群的一些发现。这项研究将系统地发展这种分解的后果。这些结果在约化p-adic群的表示理论中有着广泛的应用。更准确地说，PI计划描述共中心和轨道积分之间的关系;不同基团的共中心之间的关系，从基础变化和内部形式产生的;和共中心和mod-l表示之间的关系。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.