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

Published online by Cambridge University Press:  21 September 2020

Dmitry Gourevitch
Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, Rehovot76100, Israel e-mail: URL:
Henrik P. A. Gustafsson
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: URL:
Axel Kleinschmidt
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:
Daniel Persson*
Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
Siddhartha Sahi
Department of Mathematics, Rutgers University, Hill Center—Busch Campus, 110 Frelinghuysen Road Piscataway, NJ08854-8019, USA e-mail:
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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.

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