A reduction principle for Fourier coefficients of automorphic forms

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

Research output: Contribution to journalArticlepeer-review


We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group G(AK) , 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 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.

Original languageEnglish (US)
Pages (from-to)2679-2717
Number of pages39
JournalMathematische Zeitschrift
Issue number3
StatePublished - Mar 2022
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


  • Automorphic function
  • Automorphic representation
  • Fourier expansion on covers of reductive groups
  • Nilpotent orbit
  • Wave-front set
  • Whittaker support


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