An Identity relating Eisenstein series on general linear groups

TL;DR

This paper establishes a broad identity connecting Eisenstein series on general linear groups, involving constructions with parabolic subgroups and representations, and relates it to zeta functions and previous methods.

Contribution

It introduces a new identity for Eisenstein series on GL(n), generalizing the doubling method and linking it to Godement-Jacquet zeta functions.

Findings

Proves convergence and meromorphic continuation of local integrals.

Expresses Eisenstein series in terms of degenerate series.

Unramified calculation yields Godement-Jacquet zeta function.

Abstract

We give a general identity relating Eisenstein series on general linear groups. We do it by constructing an Eisenstein series, attached to a maximal parabolic subgroup and a pair of representations, one cuspidal and the other a character, and express it in terms of a degenerate Eisenstein series. In the local fields analogue, we prove the convergence in a half plane of the local integrals, and their meromorphic continuation. In addition, we find that the unramified calculation gives the Godement-Jacquet zeta function. This realizes and generalizes the construction proposed by Ginzburg and Soudry in Section 3 in their aritcle "Integral derived from the doubling method".