Geodesic
Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups
Canadian journal of mathematics, Tome 74 (2022) no. 1, pp. 122-169

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DOI

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.
DOI : 10.4153/S0008414X20000711
Mots-clés : Automorphic function, small representations, minimal representation, next-to-minimal representation, Fourier coefficient, Whittaker coefficient, Whittaker support, nilpotent orbit, wave-front set, string theory 
Gourevitch, Dmitry; Gustafsson, Henrik P. A.; Kleinschmidt, Axel; Persson, Daniel; Sahi, Siddhartha. Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups. Canadian journal of mathematics, Tome 74 (2022) no. 1, pp. 122-169. doi: 10.4153/S0008414X20000711
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     title = {Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups},
     journal = {Canadian journal of mathematics},
     pages = {122--169},
     year = {2022},
     volume = {74},
     number = {1},
     doi = {10.4153/S0008414X20000711},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X20000711/}
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