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On Fourier coefficients of automorphic forms of symplectic groups. (English) Zbl 1027.11034

For a number of reasons it is interesting to determine Fourier coefficients of automorphic forms. The best known Fourier coefficient is the so-called Whittaker Fourier coefficient. While every cuspidal representation of \(\text{GL}_n(\mathbb A)\) has such a Fourier coefficient, for other classical groups this is not true. In the paper under the review, the authors study Fourier coefficients of cuspidal representations on symplectic groups. They associate to each unipotent orbit a set of Fourier coefficients of an automorphic form on \(\text{Sp}_{2n}\) or its double cover. They prove that every cuspidal representation on a symplectic group has a non-trivial Fourier coefficient with respect to a certain type of unipotent class.

MSC:

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
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