Gelfand pairs and gamma factors mod ℓ

Speaker

Robin Zhang

Massachusetts Institute of Technology

I am an NSF Postdoctoral Fellow at MIT. I received my PhD from Columbia University under the supervision of Michael Harris in May 2023.

My main research interests are in number theory, automorphic representations, and Diophantine geometry. In particular:

• Special values of L-functions

• Restriction problems

• Rational, torsion, and periodic points

Abstract

I will discuss two uniqueness phenomena in the representation theory of finite and compact groups. First, we have the classical complex theory of Gelfand pairs and its generalizations; this has many applications to number theory and automorphic forms, such as the uniqueness of Whittaker models and the non-vanishing of the central value of triple product L-functions. Second, we have the local converse theorems that say that irreducible representations have unique Rankin–Selberg gamma factors; in fact, the local Langlands correspondence for GL(n) can be uniquely characterized by gamma factors. We will show how both phenomena can be extended to positive characteristic using two techniques involving module projectivity. The second part of this talk is based on joint work with J. Bakeberg, M. Gerbelli-Gauthier, H. Goodson, A. Iyengar, and G. Moss.