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

Abstract

Fourier coefficients of automorphic representations $\pi$ of $\Sp_{2n}(\BA)$ are attached to unipotent adjoint orbits in $\Sp_{2n}(F)$, where $F$ is a number 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 coefficients of $\pi$ are special, and prove, under a well acceptable assumption, that if $\pi$ is cuspidal, then the stabilizer attached to each of those maximal unipotent orbits is $F$-anisotropic as algebraic group over $F$. These results strengthen, refine and extend the earlier work of Ginzburg, Rallis and Soudry on the subject. As a consequence, we obtain constraints on those maximal unipotent orbits if $F$ is totally imaginary, further applications of which to the discrete spectrum with the Arthur classification will be considered in our future work.

arXiv comment48 pages. Accepted by Journal of Number Theory (Mar. 2014)