Higher weight on GL(3), II: The cusp forms

Abstract

The purpose of this paper is to collect, extend, and make explicit the results of Gel’fand, Graev and Piatetski-Shapiro and Miyazaki for the $GL\left(3\right)$ cusp forms which are nontrivial on $SO\left(3,ℝ\right)$. 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 $L^2\left(SL\left(3,\mathbb{Z}\right)\setminus PSL\left(3,ℝ\right)\right)$, accounting for multiplicities, in the style of Duke, Friedlander and Iwaniec’s paper. We do this at a level of uniformity suitable for Poincaré series which are not necessarily $K$-finite. We directly compute the Jacquet integral for the Whittaker functions at the minimal $K$-type, improving Miyazaki’s computation. These results will form the basis of the nonspherical spectral Kuznetsov formulas and the arithmetic/geometric Kuznetsov formulas on $GL\left(3\right)$. The primary tool will be the study of the differential operators coming from the Lie algebra on vector-valued cusp forms.