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

abstract: Let F be a totally real number field and n 3. Let and be cuspidal automorphic representations for PGL₍+₁ (F) and PGL₍-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, ) and L (1/2, ) for certain sequences of 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, ) L (1/2, ) over a conductor aspect family of.