The arithmetic Kuznetsov formula on $GL(3)$, I: The Whittaker case

TL;DR

This paper develops a new arithmetic Kuznetsov formula for $SL(3,\mathbb{Z})$ focusing on the spherical $GL(3)$ Whittaker function, enabling the study of Kloosterman sums and related zeta functions.

Contribution

It introduces the first arithmetic Kuznetsov formula for $SL(3,\mathbb{Z})$ that isolates the spherical Whittaker function, advancing spectral analysis techniques.

Findings

Derived a formula isolating the spherical $GL(3)$ Whittaker function.

Applied the formula to analyze Kloosterman sums and zeta functions.

Provided new tools for studying automorphic forms on $SL(3)$.

Abstract

The original formulae of Kuznetsov for $SL(2,\mathbb{Z})$ allowed one to study either a spectral average via Kloosterman sums or to study an average of Kloosterman sums via a spectral interpretation. In previous papers, we have developed the spectral Kuznetsov formulae at the minimal weights for $SL(3,\mathbb{Z}))$, and in these formulae, the big-cell Kloosterman sums occur with weight functions attached to four different integral kernels, according to the choice of signs of the indices. These correspond to the $J$- and $K$-Bessel functions in the case of $GL(2)$. In this paper, we demonstrate a linear combination of the spherical and weight one $SL(3,\mathbb{Z})$ Kuznetsov formulae that isolates one particular integral kernel, which is the spherical $GL(3)$ Whittaker function. Using the known inversion formula of Wallach, we give the first arithmetic Kuznetsov formula for $SL(3,\mathbb{Z})$ and use it to study smooth averages and the Kloosterman zeta function attached to this particular choice of signs.