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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
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= {\rm 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

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.