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

Abstract

We prove a reciprocity formula for the average of the product of Rankin–Selberg $L$-functions $L(1/2,\mathrm{\Pi }\times \tilde{\sigma })L(1/2,\sigma \times \tilde{\pi })$ as $\sigma$ varies over automorphic representations of $\mathrm{PGL}(n)$ over a number field $F$, where $\mathrm{\Pi }$ and $\pi$ are cuspidal automorphic representations of $\mathrm{PGL}(n+1)$ and $\mathrm{PGL}(n-1)$ over $F$, respectively. If $F$ is totally real, and $\mathrm{\Pi }$ 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.