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

Buttcane, Jack

doi:10.2140/ant.2018.12.2237

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

Jack Buttcane

Vol. 12 (2018), No. 10, 2237–2294

DOI: 10.2140/ant.2018.12.2237

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 $SO (3, ℝ)$ . 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 $L^{2} (SL (3, ℤ) ∖ PSL (3, ℝ))$ , 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.

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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

Vol. 12, No. 10, 2018