Spectral reciprocity for $\mathrm{GL}(n)$ and simultaneous non-vanishing of central $L$-values

Abstract

Let $F$ be a totally real number field and $n\ge 3$. Let $Π$ and $π$ be cuspidal automorphic representations for $\mathrm{PGL}_{n+1}(F)$ and $\mathrm{PGL}_{n-1}(F)$, respectively, that are unramified and tempered at all finite places. We prove simultaneous non-vanishing of the Rankin--Selberg $L$-values $L(1/2,Π\otimes\widetildeσ)$ and $L(1/2,σ\otimes\widetildeπ)$ for certain sequences of $σ$ varying over cuspidal automorphic representations for $\mathrm{PGL}_n(F)$ with conductor tending to infinity in the level aspect and bearing certain local conditions. Along the way, we also prove a reciprocity formula for the average of the product of Rankin--Selberg $L$-functions $L(1/2,Π\otimes\widetildeσ)L(1/2,σ\otimes\widetildeπ)$ over a conductor aspect family of $σ$.