Spectral reciprocity for GL( n ) and simultaneous non-vanishing of central L -values

Let F be a totally real number field and n≥ 3. Let Π and π be cuspidal automorphic representations for PGLₙ₊₁(F) and PGLₙ₋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⊗σ) and L(1/2,σ⊗π) for certain sequences of σ varying over cuspidal automorphic representations for PGLₙ(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⊗σ)L(1/2,σ⊗π) over a conductor aspect family of σ.