Un caractère relatif pondéré

TL;DR

The article investigates the relative trace formula for the symmetric space S = GL(2p+1)/GL(p+1)×GL(p) over a number field, showing that its relatively cuspidal part is induced from the cuspidal part of the space associated with (GL(1)×GL(2p)) ace (GL(1)×GL(p)×GL(p)) and characterizing its spectral contribution.A central development is the construction of a truncated kernel Λ^T_ heta and the resulting distributions J_hi^T and J_hi, which are expressed in terms of weighted relative characters J_{P,pi} built from Eisenstein series and intertwining periods, connecting the spectral side to Langlands data.The authors prove that non-vanishing of certain χ-components corresponds to the representations σ being symplectic, linking the spectral content to Lusztig-type L-functions and Langlands parameters factoring through the dual-group morphism rho, as predicted by relative Langlands philosophy.The results provide explicit formulas and covariance properties for the relative characters, enabling a precise decomposition of the Guo-Jacquet trace formula and offering a framework to compare relative trace formulas via Eisenstein-series techniques.

Abstract

Let $p\geq 1$. The symmetric space $S=GL(2p+1)/GL(p+1)\times GL(p)$ (over a number field) is not cuspidal in the sense that its automorphic spectrum does not contain any cuspidal representation of $GL(2p+1)$. In this article, we compute the spectral decomposition of its relatively cuspidal part: this is, by definition, the part of the spectrum that is induced from the cuspidal part of the symmetric space $(GL(1)\times GL(2p)) / (GL(1)\times GL(p)\times GL(p))$. As an application, we obtain the expression of the contribution of this relatively cuspidal part to the Guo-Jacquet trace formula (established by H. Li and the author) in terms of a weighted relative character.