Bessel functions on GL(n), I

TL;DR

This paper develops the theory of Bessel functions on GL(n) within the Kuznetsov trace formula framework, proving key conjectures and providing new results for GL(n) and GL(4) cases, advancing understanding of automorphic forms.

Contribution

It proves one of the main conjectures for GL(n) and most for the long Weyl element, and advances series and integral representations for GL(4) Bessel functions.

Findings

Proved a key conjecture for GL(n) Bessel functions.

Established unconditional results on Archimedean Whittaker functions for GL(n).

Made progress on series and integral representations for GL(4).

Abstract

In the context of the Kuznetsov trace formula, we outline the theory of the Bessel functions on $GL(n)$ 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 $GL(n)$ 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 $GL(n)$ with arbitrary weight. We expect the heuristics here to apply at the level of real reductive groups. In an appendix, we make good progress toward series and integral representations of $GL(4)$ Bessel functions by proving several of the conjectures for $GL(4)$.