On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups (Q462834)

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 \textit{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 \(F\).