On Shalika periods and a theorem of Jacquet-Martin

Let $\pi$ be a cuspidal automorphic representation of $GL_4(\Bbb A)$ with central character $\mu^2$. It is known that $\pi$ has Shalika period with respect to $\mu$ if and only if the $L$-function $L^S(s, \pi, \bigwedge^2\otimes\mu^{-1})$ has a pole at $s=1$. Jacquet and Martin considered the analogous question for cuspidal representations $\pi_D$ of the inner form $GL_2(D)(\Bbb A)$, and obtained a partial result via the relative trace formula. In this paper, we provide a complete solution to this problem via the method of theta correspondence, and give necessary and sufficient conditions for the existence of Shalika period for $\pi_D$. We also resolve the analogous question in the local setting.