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# Globalization of Distinguished Supercuspidal Representations of $\GL(n)$

Published:2002-06-01
Printed: Jun 2002
• Jeffrey Hakim
• Fiona Murnaghan
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## Abstract

An irreducible supercuspidal representation $\pi$ of $G= \GL(n,F)$, where $F$ is a nonarchimedean local field of characteristic zero, is said to be distinguished'' by a subgroup $H$ of $G$ and a quasicharacter $\chi$ of $H$ if $\Hom_H(\pi,\chi)\noteq 0$. There is a suitable global analogue of this notion for and irreducible, automorphic, cuspidal representation associated to $\GL(n)$. 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.
 MSC Classifications: 22E50 - Representations of Lie and linear algebraic groups over local fields [See also 20G05] 22E35 - Analysis on $p$-adic Lie groups 11F70 - Representation-theoretic methods; automorphic representations over local and global fields

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