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
DOI:
https://doi.org/10.1090/ert/565
Published electronically:
June 7, 2021
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Abstract | References | Similar Articles | Additional Information
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
Email:
dmitry.gourevitch@weizmann.ac.il
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
Email:
gustafsson@ias.edu
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
Email:
axel.kleinschmidt@aei.mpg.de
Daniel Persson
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
MR Author ID:
799483
Email:
daniel.persson@chalmers.se
Siddhartha Sahi
Affiliation:
Department of Mathematics, Rutgers University, Hill Center - Busch Campus, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
MR Author ID:
153000
Email:
sahi@math.rugers.edu
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.
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American Mathematical Society