Un caract\`ere relatif pond\'er\'e

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.