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

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- by Jeffrey Hakim
- Represent. Theory
**17**(2013), 120-175 - DOI: https://doi.org/10.1090/S1088-4165-2013-00426-0
- Published electronically: March 4, 2013
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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 $G$.## References

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## Bibliographic Information

**Jeffrey Hakim**- Affiliation: Department of Mathematics and Statistics, American University, Washington, DC 20016
- MR Author ID: 272088
- Email: jhakim@american.edu
- 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.
- © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3027804