Speh representations are relatively discrete

Abstract

Let $F$ be a $p$-adic field of characteristic zero and odd residual characteristic. Let $\mathbf{Sp}_{2n}(F)$ denote the symplectic group defined over $F$, where $n\geq 2$. We prove that the Speh representations $\mathcal{U}(\delta,2)$, where $\delta$ is a discrete series representation of $\mathbf{GL}_n(F)$, lie in the discrete spectrum of the $p$-adic symmetric space $\mathbf{Sp}_{2n}(F) \backslash \mathbf{GL}_{2n}(F)$.