Bessel F-crystals for reductive groups

Xinwen Zhu, California Institute of Technology

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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.