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

Fourier coefficients of automorphic representations π   of Sp2n(A)Sp2n(A) are attached to unipotent adjoint orbits in Sp2n(F)Sp2n(F), where F   is a number field and AA is the ring of adeles of F. We prove that for a given π, all maximal unipotent orbits that give nonzero Fourier coefficients of π are special, and prove, under a well-acceptable assumption, that if π 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.