On linear periods

Let ${\pi }^{\mathrm{\prime }}$ be a cuspidal automorphic representation of $G{L}_{2n}$, which is assumed to be the Jacquet-Langlands transfer from a cuspidal automorphic representation $\pi$ of $G{L}_{2m}(D)$, where $D$ is a division algebra so that $G{L}_{2m}(D)$ is an inner form of $G{L}_{2n}$. In this paper, we consider the relation between linear periods on $\pi$ and ${\pi }^{\mathrm{\prime }}$. We conjecture that the non-vanishing of the linear period on $\pi$ would imply the non-vanishing of that on ${\pi }^{\mathrm{\prime }}$. We illustrate an approach using a relative trace formula towards this conjecture, and prove the existence of smooth transfer over non-archimedean local fields.