Symplectic period for a representation of $G{L}_n(D)$

Abstract

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

26 pages, comments are welcome