The Spectral Kuznetsov Formula on SL(3)

TL;DR

This paper advances the spectral Kuznetsov formula on SL(3) by representing complex weight functions as solutions to differential equations, enabling more accessible analytic techniques for automorphic form analysis.

Contribution

It provides new differential equation-based representations of weight functions in the SL(3) Kuznetsov formula, simplifying their analysis and application.

Findings

Derived power series representations of weight functions

Developed Mellin-Barnes integral formulas for minimal dimension

Facilitated classical analytic techniques for further study

Abstract

The $SL(3)$ Kuznetsov formula exists in several versions, and has been employed with some success to study automorphic forms on $SL(3)$. In each version, the weight functions on the geometric side are given by multiple integrals with complicated oscillating factors; this is the primary obstruction to its use. By describing them as solutions to systems of differential equations, we give power series and Mellin-Barnes integral representations of minimal dimension for these weight functions. This completes the role of harmonic analysis on symmetric spaces on the geometric side of the Kuznetsov formula, so that further study may be done through classical analytic techniques.