Reciprocity and non-vanishing of central L-values

A reciprocity formula usually relates certain moments of two different families of $L$-functions which apparently have no connections between them. The first such formula was due to Motohashi who related a fourth moment of Riemann zeta values on the central line with a cubic moment of certain automorphic central $L$-values for $\mathrm{GL}(2)$. In this talk we describe a higher rank reciprocity which relates mixed moments of central Rankin--Selberg $L$-values for $\mathrm{GL}(n+1) \times \mathrm{GL}(n)$ and $\mathrm{GL}(n) \times \mathrm{GL}(n-1)$. We also describe an application of this formula to prove simultaneous non-vanishing of above $L$-values at the central point. This is a joint work with Ramon Nunes.