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Distinguished tame supercuspidal representations and odd orthogonal periods


Authors: Jeffrey Hakim and Joshua Lansky
Journal: Represent. Theory 16 (2012), 276-316
MSC (2010): Primary 22E50, 11F70; Secondary 11F67, 11E08, 11E81
DOI: https://doi.org/10.1090/S1088-4165-2012-00418-6
Published electronically: June 1, 2012
MathSciNet review: 2925798
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Abstract: We further develop and simplify the general theory of distinguished tame supercuspidal representations of reductive $ p$-adic groups due to Hakim and Murnaghan, as well as the analogous theory for finite reductive groups due to Lusztig. We apply our results to study the representations of $ \mathrm {GL}_n(F)$, with $ n$ odd and $ F$ a nonarchimedean local field, that are distinguished with respect to an orthogonal group in $ n$ variables. In particular, we determine precisely when a supercuspidal representation is distinguished with respect to an orthogonal group and, if so, that the space of distinguishing linear forms has dimension one.


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Additional Information

Jeffrey Hakim
Affiliation: Department of Mathematics and Statistics, American University, 4400 Massachusetts Avenue, NW, Washington, DC 20016
Email: jhakim@american.edu

Joshua Lansky
Affiliation: Department of Mathematics and Statistics, American University, 4400 Massachusetts Avenue, NW, Washington, DC 20016
Email: lansky@american.edu

DOI: https://doi.org/10.1090/S1088-4165-2012-00418-6
Keywords: Supercuspidal representation, involution, distinguished representation, orthogonal group.
Received by editor(s): March 7, 2011
Received by editor(s) in revised form: November 23, 2011
Published electronically: June 1, 2012
Additional Notes: Both authors were supported by NSF grant DMS-0854844.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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