Relative discrete series representations for two quotients of $p$-adic ${\mathbf{G}\mathbf{L}}_n$

We provide an explicit construction of representations in the discrete spectrum of two $p$-adic symmetric spaces. We consider $$\mathbf{G}\mathbf{L}$_n(F) \times $\mathbf{G}\mathbf{L}$n(F) \backslash $\mathbf{G}\mathbf{L}${2n}(F)$and$$\mathbf{G}\mathbf{L}$_n(F) \backslash $\mathbf{G}\mathbf{L}$_n(E)$,where$E$ is a quadratic Galois extension of a nonarchimedean local field $F$ of characteristic zero and odd residual characteristic. The proof of the main result involves an application of a symmetric space version of Casselman’s Criterion for square integrability due to Kato and Takano.