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

Dmitry Gourevitch, Henrik P.A. Gustafsson, Axel Kleinschmidt, Daniel Persson, Siddhartha Sahi

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2 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)122-169
Number of pages48
JournalCanadian Journal of Mathematics
Issue number1
StatePublished - Feb 21 2022
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


  • Automorphic function
  • Fourier coefficient
  • Whittaker coefficient
  • Whittaker support
  • minimal representation
  • next-to-minimal representation
  • nilpotent orbit
  • small representations
  • string theory
  • wave-front set


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