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Fourier coefficients of automorphic representations $\pi$ of $\Sp_{2n}(\BA)$are attached to unipotent adjoint orbits in $\Sp_{2n}(F)$, where $F$ is anumber field and $\BA$ is the ring of adeles of $F$. We prove that for a given$\pi$, all maximal unipotent orbits, which gives nonzero Fourier coefficientsof $\pi$ are special, and prove, under a well acceptable assumption, that if$\pi$ is cuspidal, then the stabilizer attached to each of those maximalunipotent orbits is $F$-anisotropic as algebraic group over $F$. These resultsstrengthen, refine and extend the earlier work of Ginzburg, Rallis and Soudryon the subject. As a consequence, we obtain constraints on those maximalunipotent orbits if $F$ is totally imaginary, further applications of which tothe discrete spectrum with the Arthur classification will be considered in ourfuture work.

On Special Unipotent Orbits and Fourier Coefficients for Automorphic Forms on Symplectic Groups