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On Special Unipotent Orbits and Fourier Coefficients for Automorphic Forms on Symplectic Groups (1309.6238v3)

Published 24 Sep 2013 in math.NT and math.RT

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.

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