Symplectic period for a representation of GLn(D)

Abstract

Let n be a natural number, taking the value 3 or 4. Let D be a quaternion division algebra over a non-archimedean local field k of characteristic zero, and let ${\mathrm{Sp}}_n\left(D\right)$ be the unique non-split inner form of the symplectic group ${\mathrm{Sp}}_{2n}\left(k\right)$. This paper classifies those irreducible admissible representations of ${\mathrm{GL}}_n\left(D\right)$ that admit a symplectic period, that is, those irreducible admissible representations $(\pi ,V)$ of ${\mathrm{GL}}_n\left(D\right)$ which have a linear functional l on V such that $l\left(\pi \right(h\left)v\right)=l\left(v\right)$ for all $v\in V$ and $h\in {\mathrm{Sp}}_n\left(D\right)$. Our results also contain all unitary representations having a symplectic period, as stated in Prasad's conjecture.