A spectral expansion for the symmetric space $\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$

TL;DR

This paper establishes a spectral expansion for theta series on the symmetric space $ ext{GL}_n(E)/ ext{GL}_n(F)$, advancing the understanding of the Jacquet-Rallis trace formula through new bounds on Eisenstein series.

Contribution

It extends the spectral expansion to the case of automorphic representations for $ ext{GL}_n$, incorporating bounds on Eisenstein series near the imaginary axis.

Findings

Spectral expansion expressed as an integral of relative characters.

Extension of bounds for Eisenstein series on neighborhoods of the imaginary axis.

Key technical ingredient for the convergence and analysis of the spectral expansion.

Abstract

In this article we state and prove the spectral expansion of theta series attached to the symmetric space $\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$ where $n\geq 1$ and $E/F$ is a quadratic extension of number fields. This is an important step towards the fine spectral expansion of the Jacquet-Rallis trace formula for general linear groups. To obtain our result, we extend the work of Jacquet-Lapid-Rogawski on intertwining periods to the case of discrete automorphic representations. The expansion we get is an absolutely convergent integral of relative characters built upon Eisenstein series and intertwining periods. We also establish a crucial but technical ingredient whose interest lies beyond the focus of the article: we prove bounds for discrete Eisenstein series of $\mathrm{GL}_n$ on a neighborhood of the imaginary axis extending previous works of Lapid on cuspidal Eisenstein series. We even need a variant of such bounds on some shifts of the imaginary axis.