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
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(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.

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