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On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups. (English) Zbl 1386.11072

Let \(F\) denote a number field, and let \(\mathbb{A}\) stand for the ring of adeles of \(F\). In the paper under the review, authors build on the previous work of D. Ginzburg et al. [Manuscr. Math. 111, No. 1, 1–16 (2003; Zbl 1027.11034); The descent map from automorphic representations of \(\text{GL}(n)\) to classical groups. Hackensack, NJ: World Scientific (2011; Zbl 1233.11056)] to obtain refined properties of Fourier coefficients of automorphic forms of the symplectic group \( \mathrm{Sp}_{2n}(\mathbb{A})\). The authors prove that all maximal unipotent orbits that give nonzero Fourier coefficients of an automorphic representation \(\pi\) of \(\mathrm{Sp}_{2n}(\mathbb{A})\) are special. It is also proved, under a certain assumption, that, for cuspidal \(\pi\), the stabilizer attached to such a maximal unipotent orbit is \(F\)-anisotropic. These results imply certain constraints on such maximal unipotent orbits for totally imaginary field \(F\).

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
11F30 Fourier coefficients of automorphic forms
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