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

Abstract

Let $D$ be a quaternion division algebra over a non-archimedean local field $K$ of characteristic zero, and let $Sp_n(D)$ be the unique non-split inner form of the symplectic group $Sp_{2n}(K)$. This paper classifies the irreducible admissible representations of $GL_{n}(D)$ with a symplectic period for $n = 3$ and $4$, i.e., those irreducible admissible representations $(π, V)$ of $GL_{n}(D)$ which have a linear functional $l$ on $V$ such that $l(π(h)v) = l(v)$ for all $v \in V$ and $h \in Sp_n(D)$. Our results also contain all unitary representations having a symplectic period, as stated in Prasad's conjecture.