Top Fourier coefficients of residual Eisenstein series on symplectic or metaplectic groups, induced from Speh representations

We consider the residues at the poles in the half plane Re(s) ≥ 0 of Eisenstein series, on symplectic groups, or their double covers, induced from Speh representations. We show that for each such pole, there is a unique maximal nilpotent orbit, attached to Fourier coefficients admitted by the corresponding residual representation. We find this orbit in each case.