The poles and residues of Eisenstein series induced from Speh representations

David Ginzburg, David Soudry

Published 2020-12-07Version 1

We consider Eisenstein series, on split special orthogonal groups, symplectic groups, or their double covers, induced from Speh representations. Except the metaplectic groups case, their poles were determined by Jiang, Liu, Zhang [JLZ13]. We give another proof for the existence of these poles, which is straightforward and works for double covers of symplectic groups, as well. In case of symplectic groups, or their double covers, we use the same proof to show that for each 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.