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

We provide an explicit construction of representations in the discrete spectrum of two $p$-adic symmetric spaces. We consider $\mathbf{GL}_n(F) \times \mathbf{GL}_n(F) \backslash \mathbf{GL}_{2n}(F)$ and $\mathbf{GL}_n(F) \backslash \mathbf{GL}_n(E)$, 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.
 Keywords: $p$-adic symmetric space, relative discrete series, Casselman’s Criterion
 MSC Classifications: 22E50 - Representations of Lie and linear algebraic groups over local fields [See also 20G05] 22E35 - Analysis on $p$-adic Lie groups