# Kuznetsov, Petersson and Weyl on GL(3), I: The principal series forms

## TL;DR

This paper develops the spectral Kuznetsov formula for non-spherical Maass forms on GL(3), enabling the study of new automorphic forms and exponential sums, and introduces a minimal method for deriving such formulae.

## Contribution

It provides the first proof of non-spherical principal series forms on GL(3) and generalizes Stade's formula and Kontorovich-Lebedev inversion methods.

## Key findings

- Derived spectral Kuznetsov formula for non-spherical Maass forms on GL(3)
- Produced generalizations of Stade's formula and Kontorovich-Lebedev inversion
- Established existence of new automorphic forms not from symmetric-square construction

## Abstract

The Kuznetsov and Petersson trace formulae for $GL(2)$ forms may collectively be derived from Poincar\'e series in the space of Maass forms with weight. Having already developed the spherical spectral Kuznetsov formula for $GL(3)$, the goal of this series of papers is to derive the spectral Kuznetsov formulae for non-spherical Maass forms and use them to produce the corresponding Weyl laws; this appears to be the first proof of the existence of such forms not coming from the symmetric-square construction. Aside from general interest in new types of automorphic forms, this is a necessary step in the development of a theory of exponential sums on $GL(3)$. We take the opportunity to demonstrate a sort of minimal method for developing Kuznetsov-type formulae, and produce auxillary results in the form of generalizations of Stade's formula and Kontorovich-Lebedev inversion. This first paper is limited to the non-spherical prinicpal series forms as there are some significant technical details associated with the generalized principal series forms, which will be handled in a separate paper. The best analog of this type of form on $GL(2)$ is the forms of weight one which sometimes occur on congruence subgroups.