Eulerianity of Fourier coefficients of automorphic forms. (English) Zbl 1482.11071
Constructing autormorphic \(L\)-functions is a major theme in the Langlands program. One systematic fruitful way to construct automorphic \(L\)-functions that has largely shaped the Langlands program in its current form is initiated by R. P. Langlands in his pioneering work on Eisenstein series [Euler products. Yale Mathematical Monographs. 1. New Haven-London: Yale University Press. (1971; Zbl 0231.20016)] and its constant terms, and later extended by
F. Shahidi in his work on Whittaker Fourier coefficients [Am. J. Math. 103, 297–355 (1981; Zbl 0467.12013)], i.e., the profound Langlands-Shahidi theory.
With the goal of constructing \(L\)-functions via investigating other Fourier coefficients, in the paper under review, the authors study the Eulerianity property of certain Fourier coefficients of minimal and next-to-minimal automorphic representations. Some structural results are observed, and some special examples are analyzed. However, all of those seems a little bit “trivial” and may not have “non-trivial” applications in the goal mentioned here, though most of the authors were working on string theory and applications may be seen there.
With the goal of constructing \(L\)-functions via investigating other Fourier coefficients, in the paper under review, the authors study the Eulerianity property of certain Fourier coefficients of minimal and next-to-minimal automorphic representations. Some structural results are observed, and some special examples are analyzed. However, all of those seems a little bit “trivial” and may not have “non-trivial” applications in the goal mentioned here, though most of the authors were working on string theory and applications may be seen there.
Reviewer: Caihua Luo (Göteborg)
MSC:
| 11F30 | Fourier coefficients of automorphic forms |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
| 20G45 | Applications of linear algebraic groups to the sciences |
Keywords:
Euler product; Fourier coefficients on reductive groups; Fourier-Jacobi coefficients; automorphic forms; automorphic representation; minimal representation; next-to-minimal representation; Whittaker support; nilpotent orbit; wave-front set; Eisenstein seriesReferences:
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