@article{2bd7358e766c4647bdd8ddb4bca3c800,
title = "Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups",
abstract = "In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let be a minimal or next-to-minimal automorphic representation of G. We prove that any 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. 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 and with a view toward applications to scattering amplitudes in string theory.",
keywords = "Automorphic function, Fourier coefficient, Whittaker coefficient, Whittaker support, minimal representation, next-to-minimal representation, nilpotent orbit, small representations, string theory, wave-front set",
author = "Dmitry Gourevitch and Gustafsson, {Henrik P.A.} and Axel Kleinschmidt and Daniel Persson and Siddhartha Sahi",
note = "Funding Information: We are grateful to Guillaume Bossard, Ben Brubaker, Daniel Bump, Solomon Friedberg, David Ginzburg, Joseph Hundley, Stephen D. Miller, Manish Patnaik, Boris Pioline, and Gordan Savin for helpful discussions. We thank the referees for helpful comments on an earlier version of the manuscript. We also thank the Banff International Research Station for Mathematical Innovation and Discovery and the Simons Center for Geometry and Physics for their hospitality during different stages of this project. D.G. was partially supported by ERC StG grant 637912 and BSF grant 2019274. During his time at Stanford University (when the majority of this paper was written), H.G. was supported by the Knut and Alice Wallenberg Foundation. Later, H.G. was supported by the Swedish Research Council (Vetenskapsr det), grant number 2018-06774. D.P. was supported by the Swedish Research Council (Vetenskapsr det), grant number 2018-04760. S.S. was partially supported by Simons Foundation grant 509766, NSF grant DMS-2001537 and BSF grant 2019724. Publisher Copyright: {\textcopyright} Canadian Mathematical Society 2020.",
year = "2022",
month = feb,
day = "21",
doi = "10.4153/S0008414X20000711",
language = "English (US)",
volume = "74",
pages = "122--169",
journal = "Canadian Journal of Mathematics",
issn = "0008-414X",
publisher = "Canadian Mathematical Society",
number = "1",
}