Non-Siegel Eisenstein series for symplectic groups

Abstract

We explicitly (in terms of Langlands parameters) describe the image of the degenerate Eisenstein series in the case of a symplectic group. We study these series for any maximal parabolic subgroup and any Grossencharacter. We do that by an explicit analysis of the constant term of the Eisenstein series. Using already available information on the constituents of the local degenerate principal series, we further analyze the action of the local intertwining operators and determine their images. The local components of the images are given explicitly, in terms of their Langlands parameters. In this way, we have a complete description of the images of Eisenstein series in the right half-plane, for any Grossencharacter of the adeles over \(\mathbb {Q}.\) Thus, we complete previously known results obtained by different techniques for the trivial character and extend them to the case of the non-trivial characters.