Archimedean period relations for Rankin-Selberg convolutions

Abstract

We formulate and prove the archimedean period relations for Rankin-Selberg convolutions of $\mathrm{GL}\left(n\right)\times \mathrm{GL}\left(n\right)$ and $\mathrm{GL}\left(n\right)\times \mathrm{GL}(n-1)$, for all generic cohomological representations. As a consequence, we prove the non-vanishing of the archimedean modular symbols. This extends the earlier results in [14] for essentially tempered representations of $\mathrm{GL}\left(n\right)\times \mathrm{GL}(n-1)$.