Spectral Reciprocity for GL$(n)$ and Simultaneous Non-Vanishing of Central $L$-Values


We prove a reciprocity formula for the average of the product of Rankin–Selberg $L$-functions $L(1/2, Pi times widetilde sigma)L(1/2, sigma times widetilde pi)$ as $ sigma$ varies over automorphic representations of $ mathrmPGL(n)$ over a number field $F$, where $ Pi$ and $ pi$ are cuspidal automorphic representations of $ mathrmPGL(n+1)$ and $ mathrmPGL(n-1)$ over $F$, respectively. If $F$ is totally real, and $ Pi$ and $ pi$ are tempered everywhere, we deduce simultaneous non-vanishing of these $L$-values for certain sequences of $ sigma$ with conductor tending to infinity in the level aspect and bearing certain local conditions.