Cocenters and Representations of Reductive p-adic Groups

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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.