The fine spectral expansion of the Rankin-Selberg period

We state and prove the spectral expansion of the theta series attached to the Rankin-Selberg spherical variety $(\mathrm{GL}_{n+1} \times \mathrm{GL}_n)/\mathrm{GL}_n$. This is a key result towards the fine spectral expansion of the Jacquet-Rallis trace formula. Our expansion is written in terms of regularized Rankin--Selberg periods for non-tempered automorphic representations, which we show compute special values of $L$-functions. The proof relies on shifts of contours of integration à la Langlands. We also establish two technical but crucial results on bounds and singularities for discrete Eisenstein series of $\mathrm{GL}_n$ in the positive Weyl chamber.