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Eulerianity of Fourier coefficients of automorphic forms

Authors: Dmitry Gourevitch, Henrik P. A. Gustafsson, Axel Kleinschmidt, Daniel Persson and Siddhartha Sahi
Journal: Represent. Theory 25 (2021), 481-507
MSC (2020): Primary 11F30, 11F70, 22E55, 20G45
Published electronically: June 7, 2021
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Abstract: We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a ‘hidden’ invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier–Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class of Eisenstein series in minimal and next-to-minimal representations of groups of ADE-type that are of interest in string theory.

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Additional Information

Dmitry Gourevitch
Affiliation: Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel
MR Author ID: 843930
ORCID: 0000-0001-6436-2092

Henrik P. A. Gustafsson
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540; Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854; Department of Mathematical Sciences, University of Gothenburg; and Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
MR Author ID: 1157059
ORCID: 0000-0002-3364-1547

Axel Kleinschmidt
Affiliation: Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Am Mühlen- berg 1, DE-14476 Potsdam, Germany; and International Solvay Institutes, ULB-Campus Plaine CP231, BE-1050, Brussels, Belgium
MR Author ID: 721044

Daniel Persson
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
MR Author ID: 799483

Siddhartha Sahi
Affiliation: Department of Mathematics, Rutgers University, Hill Center - Busch Campus, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
MR Author ID: 153000

Keywords: Euler product, Fourier coefficients on reductive groups, Fourier–Jacobi coefficients, automorphic forms, automorphic representation, minimal representation, next-to-minimal representation, Whittaker support, nilpotent orbit, wave-front set, Eisenstein series.
Received by editor(s): May 12, 2020
Received by editor(s) in revised form: October 7, 2020
Published electronically: June 7, 2021
Additional Notes: The first author was partially supported by ERC StG grant 637912 and BSF grant 2019724. The second and fourth authors were supported by the Swedish Research Council (Vetenskapsrådet), grants 2018-06774 and 2018-04760, respectively. The fifth author was partially supported by NSF grants DMS-1939600 and DMS-2001537, and Simons’ foundation grant 509766.
Article copyright: © Copyright 2021 American Mathematical Society