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Sorry if this is a vague question. I remember from my younger days that before proving his trace formula, Kuznetsov had a pretty result on orthogonality of Bessel functions. The formulas that I am ...

It is a convention to say Kloosterman sums and Bessel functions are similar. There are papers talking about Bessel functions on $p$-adic group (associated with a representation) such as Baruch's: ...