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Globalization of Distinguished Supercuspidal Representations of GL(n)
Published online by Cambridge University Press: 20 November 2018
Abstract
An irreducible supercuspidal representation $\pi$ of
$G\,=\,\text{GL}\left( n,\,F \right)$, where
$F$ is a nonarchimedean local field of characteristic zero, is said to be “distinguished” by a subgroup
$H$ of
$G$ and a quasicharacter
$\text{ }\!\!\chi\!\!\text{ }$ of
$H$ if
$\text{Ho}{{\text{m}}_{H}}\left( \pi ,\,\text{ }\!\!\chi\!\!\text{ } \right)\,\ne \,0$. There is a suitable global analogue of this notion for an irreducible, automorphic, cuspidal representation associated to
$\text{GL}\left( n \right)$. Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.
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- Copyright © Canadian Mathematical Society 2002
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