Vol. 12, No. 10, 2018

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 7, 1239–1357
Issue 6, 1127–1237
Issue 5, 981–1126
Issue 4, 805–980
Issue 3, 541–804
Issue 2, 267–539
Issue 1, 1–266

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
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