The fine spectral expansion of the Rankin-Selberg period

Abstract

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.