Tempered Symplectic Distinction For Quaternionic $G{L}_n(D)$: A Conjecture of PRASAD

Let $F$ be a non-Archimedean local field of characteristic zero, let $D\mathrm{/}F$ be the quaternion division algebra, and put $G_n=G{L}_n(D)$ and $H_n=S{p}_n(D)$. We prove Prasad’s revised conjecture on tempered representations with symplectic period. If $\rho$ is a unitary $H_r$-distinguished supercuspidal representation and ${\delta }_m(\rho )$ denotes the generalized Steinberg representation associated to $\rho$, set $B_0(\rho )=\rho ,B_m(\rho )={\delta }_{m+1}(\rho )\times {\delta }_m(\rho ),m\ge 1.$ We prove that an irreducible tempered representation $\pi$ of $G_n$ is $H_n$-distinguished if and only if $\pi$ is isomorphic to the product over $i$ of $B_{m_i}({\rho }_i),$ where each ${\rho }_i$ is a unitary distinguished supercuspidal representation.