Vol. 12, No. 10, 2018

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

Jack Buttcane

Vol. 12 (2018), No. 10, 2237–2294

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(3) cusp forms which are nontrivial on SO(3, ). 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 L2(SL(3, )PSL(3, )), 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(3). The primary tool will be the study of the differential operators coming from the Lie algebra on vector-valued cusp forms.

Maass forms, automorphic forms, GL(3), SO(3), weight, raising and lowering operators
Mathematical Subject Classification 2010
Primary: 11F72
Secondary: 11F30
Received: 9 April 2017
Revised: 15 April 2018
Accepted: 19 August 2018
Published: 1 February 2019
Jack Buttcane
Mathematics Department
University at Buffalo
Buffalo, NY
United States