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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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A reduction principle for Fourier coefficients of automorphic forms
Abstract We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients.
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- Award ID(s):
- 2001537
- PAR ID:
- 10332023
- Date Published:
- Journal Name:
- Mathematische Zeitschrift
- Volume:
- 300
- Issue:
- 3
- ISSN:
- 0025-5874
- Page Range / eLocation ID:
- 2679 to 2717
- 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/10.1007/s00209-021-02784-w
