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

TL;DR

This paper investigates the Fourier coefficients of residual Eisenstein series on symplectic and metaplectic groups induced from Speh representations, identifying the associated maximal nilpotent orbits for each pole.

Contribution

It establishes a unique correspondence between poles of Eisenstein series and maximal nilpotent orbits in residual representations, providing explicit orbit descriptions.

Findings

Identified the maximal nilpotent orbit for each pole of Eisenstein series.

Connected poles of Eisenstein series to specific Fourier coefficients and nilpotent orbits.

Enhanced understanding of residual representations on symplectic and metaplectic groups.

Abstract

We consider the residues at the poles in the right half plane 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.