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TL;DR

This paper computes the spectral decomposition of the relatively cuspidal spectrum of a specific symmetric space, providing explicit formulas for its contribution to the Guo-Jacquet trace formula.

Contribution

It explicitly determines the spectral decomposition of the relatively cuspidal part of a symmetric space and expresses its contribution to the trace formula in terms of weighted relative characters.

Findings

Spectral decomposition of the relatively cuspidal spectrum computed.

Explicit expression for contribution to the Guo-Jacquet trace formula obtained.

Provides tools for further analysis of automorphic spectra on symmetric spaces.

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.