Bessel functions on GL(n), I

Abstract

In the context of the Kuznetsov trace formula, we outline the theory of the Bessel functions on $GL\left(n\right)$ as a series of conjectures designed as a blueprint for the construction of Kuznetsov-type formulas with given ramification at infinity. We are able to prove one of the conjectures at full generality on $GL\left(n\right)$ and most of the conjectures in the particular case of the long Weyl element; as with previous papers, we give some unconditional results on Archimedean Whittaker functions, now on $GL\left(n\right)$ with arbitrary weight. We expect the heuristics here to apply at the level of real reductive groups. A forthcoming paper will address the initial conjectures up to Mellin-Barnes integral representations in the case of $GL\left(4\right)$ Bessel functions.