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Relative Discrete Series Representations for Two Quotients of $p$-adic $\mathbf{GL}_n$

  Published:2018-03-13
 Printed: Dec 2018
  • Jerrod Manford Smith,
    Department of Mathematics, University of Toronto, Toronto, Canada
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Abstract

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 $p$-adic symmetric space, relative discrete series, Casselman’s Criterion
MSC Classifications: 22E50, 22E35 show english descriptions Representations of Lie and linear algebraic groups over local fields [See also 20G05]
Analysis on $p$-adic Lie groups
22E50 - Representations of Lie and linear algebraic groups over local fields [See also 20G05]
22E35 - Analysis on $p$-adic Lie groups
 

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