Bessel F-crystals for reductive groups

Xinwen Zhu
California Institute of Technology
Thursday, March 4, 2021 - 3:30pm to 4:30pm
via Zoom ID 91514486597 (contact faculty for pw)

I will first review the relationship between the classical Bessel differential equation

z^2f''(z)+zf'(z)+zf(z)=0

and the classical Kloosterman sum

\sum_{x=1}^{p-1} e((x+x*)/p), where e(-)=exp(2\pi i -) and x* is the inverse of x mod p

following the work of Deligne, Dwork and Katz. Then I will discuss a generalization of this story from the point of view of Langlands duality, based on the works by Frenkel-Gross, Heinloth-Ngo-Yun, myself, and the recent joint work with Daxin Xu. In particular, the joint work with Xu gives (probably) the first example of a p-adic version of the geometric Langlands correspondence. It allows us to prove a conjecture of Heinloth-Ngo-Yun on the functoriality of some specific automorphism forms.