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On linear periods

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