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

TL;DR

This paper establishes an absolutely convergent spectral expansion for theta series on the symmetric space $GL_E(n)/GL_F(n)$ by expressing the Flicker–Rallis kernel as a sum of relative characters $\mathcal{J}_{L,\pi}(g,f,\lambda)$ integrated over unitary spectra associated to discrete automorphic representations. The core technical achievement extends Jacquet–Lapid–Rogawski’s framework from cuspidal to discrete Eisenstein data, providing sharp bounds near the imaginary axis and holomorphy properties for normalized intertwining operators. A crucial analytic tool is the mixed truncation approach of Jacquet–Lapid–Rogawski, which links truncated Flicker–Rallis periods to regularized Eisenstein periods and enables a precise spectral decomposition compatible with Jacquet–Ral lis-type trace formulas. The results advance the fine spectral expansion of the Jacquet–Ral lis trace formula for general linear groups and have broader implications for relative trace formulas and distinguished representations in the Langlands program.

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.