A spectral expansion for the symmetric space $\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$
Abstract
In this article we state and prove the spectral expansion of theta series attached to the symmetric space where and 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 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.
Keywords
Cite
@article{arxiv.2204.06069,
title = {A spectral expansion for the symmetric space $\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$},
author = {Pierre-Henri Chaudouard},
journal= {arXiv preprint arXiv:2204.06069},
year = {2025}
}
Comments
In the previous version, a shift was overlooked in the scalar product of truncated Eisenstein series and in the definition of regularized periods.The statements have been corrected. This implies, at various points in the proof, to control the Eisenstein series away from the imaginary axis. This is done in the completely rewritten section 3. The main spectral decomposition remains unchanged