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Speh representations are relatively discrete


Author: Jerrod Manford Smith
Journal: Represent. Theory 24 (2020), 525-550
MSC (2010): Primary 22E50; Secondary 22E35
DOI: https://doi.org/10.1090/ert/550
Published electronically: October 27, 2020
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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)$.


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Additional Information

Jerrod Manford Smith
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada, T2N 1N4
Email: jerrod.smith@ucalgary.ca

DOI: https://doi.org/10.1090/ert/550
Keywords: Relative discrete series, distinguished representation, symplectic group, Speh representation
Received by editor(s): August 3, 2018
Received by editor(s) in revised form: July 9, 2020
Published electronically: October 27, 2020
Article copyright: © Copyright 2020 American Mathematical Society