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In this paper, we study the irreducible admissible representations of$GL_{n}(D)$ with a symplectic period for $n = 3$ and $4$, i.e., thoserepresentations which have a linear functional invariant under $Sp_n(D)$, where$D$ is a quaternion division algebra over a non-archimedean local field $k$ ofcharacteristic zero and $Sp_n(D)$ is the unique non-split inner form of thesymplectic group $Sp_{2n}(k)$. Our results contain all distinguished unitaryrepresentations stated in Prasad's conjecture.

Symplectic period for a representation of $GL_n(D)$