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Abstract 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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Eulerianity of Fourier coefficients of automorphic forms
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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- PAR ID:
- 10339420
- Date Published:
- Journal Name:
- Representation Theory of the American Mathematical Society
- Volume:
- 25
- Issue:
- 16
- ISSN:
- 1088-4165
- Page Range / eLocation ID:
- 481 to 507
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/ert/565
