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 cusp forms which are nontrivial on . We give new descriptions of the spaces of cusp forms of minimal -type and from the Fourier–Whittaker expansions of such forms give a complete and completely explicit spectral expansion for , 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 -finite. We directly compute the Jacquet integral for the Whittaker functions at the minimal -type, improving Miyazaki’s computation. These results will form the basis of the nonspherical spectral Kuznetsov formulas and the arithmetic/geometric Kuznetsov formulas on . The primary tool will be the study of the differential operators coming from the Lie algebra on vector-valued cusp forms.
"Higher weight on GL(3), II: The cusp forms." Algebra Number Theory 12 (10) 2237 - 2294, 2018. https://doi.org/10.2140/ant.2018.12.2237