Speh representations are relatively discrete

Abstract

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