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Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

Part of: Lie groups Discontinuous groups and automorphic forms Linear algebraic groups and related topics

Published online by Cambridge University Press:  21 September 2020

Dmitry Gourevitch ,
Henrik P. A. Gustafsson ,
Axel Kleinschmidt ,
Daniel Persson  and
Siddhartha Sahi
Dmitry Gourevitch
Affiliation:
Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, Rehovot76100, Israel e-mail: dmitry.gourevitch@weizmann.ac.il URL: http://www.wisdom.weizmann.ac.il/~dimagur
Henrik P. A. Gustafsson
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA94305-2125, USA [left September 2019] and School of Mathematics, Institute for Advanced Study, Princeton, NJ08540, USA and Department of Mathematics, Rutgers University, Piscataway, NJ08854, USA and Department of Mathematical Sciences, University of Gothenburg and Chalmers University of Technology, GothenburgSE-412 96, Sweden e-mail: gustafsson@ias.edu URL: http://hgustafsson.se
Axel Kleinschmidt
Affiliation:
Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg 1, DE-14476 Potsdam, GermanyInternational Solvay Institutes, ULB-Campus Plaine CP231, BE-1050, Brussels, Belgium e-mail: axel.kleinschmidt@aei.mpg.de
Daniel Persson*
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
Siddhartha Sahi
Affiliation:
Department of Mathematics, Rutgers University, Hill Center—Busch Campus, 110 Frelinghuysen Road Piscataway, NJ08854-8019, USA e-mail: sahi@math.rugers.edu
*
e-mail: daniel.persson@chalmers.se
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Abstract

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In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let $\pi $ be a minimal or next-to-minimal automorphic representation of G. We prove that any $\eta \in \pi $ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on $\operatorname {GL}_n$ . We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type $D_5$ and $E_8$ with a view toward applications to scattering amplitudes in string theory.

Keywords

Automorphic functionsmall representationsminimal representationnext-to-minimal representationFourier coefficientWhittaker coefficientWhittaker supportnilpotent orbitwave-front setstring theory

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings 20G45: Applications to physics
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 1 , February 2022 , pp. 122 - 169
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2020

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