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Tame supercuspidal representations of $\mathrm {GL}_n$ distinguished by orthogonal involutions


Author: Jeffrey Hakim
Journal: Represent. Theory 17 (2013), 120-175
MSC (2010): Primary 22E50, 11F70; Secondary 11F67, 11E08, 11E81
DOI: https://doi.org/10.1090/S1088-4165-2013-00426-0
Published electronically: March 4, 2013
MathSciNet review: 3027804
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Abstract: For a $p$-adic field $F$ of characteristic zero, the embeddings of a tame supercuspidal representation $\pi$ of $G= \textrm {GL}_n (F)$ in the space of smooth functions on the set of symmetric matrices in $G$ are determined. It is shown that the space of such embeddings is nonzero precisely when $-1$ is in the kernel of $\pi$ and, in this case, this space has dimension four. In addition, the space of $H$-invariant linear forms on the space of $\pi$ is determined whenever $H$ is an orthogonal group in $n$ variables contained in $G$.


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Additional Information

Jeffrey Hakim
Affiliation: Department of Mathematics and Statistics, American University, Washington, DC 20016
MR Author ID: 272088
Email: jhakim@american.edu

Keywords: Supercuspidal representation, involution, distinguished representation, orthogonal group
Received by editor(s): August 16, 2011
Received by editor(s) in revised form: May 11, 2012, July 22, 2012, July 25, 2012, and September 11, 2012
Published electronically: March 4, 2013
Additional Notes: The author was supported by NSF grant DMS-0854844.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.