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(The colloquium passwordÌýwill be distributed to Princeton University and IAS members. We ask that youÌýdo not share this password. If you would like to be included in the colloquiumÌýand are not a member of either institution, please email the organizer Casey Kelleher (caseyk@princeton.edu) with an email requesting to participate which introduces yourself, your current affiliation and stage in your career.)

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I will first review the relationship between the classical Bessel differential equationÌýz^2f’’(z)+zf’(z)+zf(z)=0Ìýand the classical Kloosterman sum\sum_{x=1}^{p-1} Ìýe((x+ x*)/p), Ìý Ìý where e(-)=exp(2\pi i -) and x* is the inverse of x mod pÌýfollowing the work of Deligne, Dwork and Katz.ÌýThen 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,Ìýand 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.