# Higher Weight on GL(3), II - The cusp forms

## TL;DR

This paper explicitly describes the spectral decomposition of $GL(3)$ cusp forms, including Fourier-Whittaker expansions and Jacquet integrals, advancing understanding of automorphic forms on $SL(3,bZ)\backslash PSL(3,\bbR)$.

## Contribution

It provides new explicit descriptions of $GL(3)$ cusp forms, including spectral expansion and improved Jacquet integral computations, building on prior foundational work.

## Key findings

- Complete spectral expansion for $L^2(SL(3,\mathbb{Z})\backslash PSL(3,\mathbb{R}))$
- Explicit Fourier-Whittaker expansions of cusp forms
- Improved computation of Jacquet integrals at minimal $K$-type

## Abstract

The purpose of this paper is to collect and make explicit the results of Gel'fand, Graev and Piatetski-Shapiro and Miyazaki for the $GL(3)$ cusp forms which are non-trivial on $SO(3,\mathbb{R})$. We give new descriptions of the spaces of cusp forms of minimal $K$-type and from the Fourier-Whittaker expansions of such forms give a complete and completely explicit spectral expansion for $L^2(SL(3,\mathbb{Z})\backslash PSL(3,\mathbb{R}))$, accounting for multiplicities, in the style of Duke, Friedlander and Iwaniec's paper on Artin $L$-functions. We directly compute the Jacquet integral for the Whittaker functions at the minimal $K$-type, improving Miyazaki's computation. The primary tool will be the study of the differential operators coming from the Lie algebra on vector-valued cusp forms.