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Self-dual representations of $ \operatorname{SL}(n,F)$


Author: Kumar Balasubramanian
Journal: Proc. Amer. Math. Soc. 144 (2016), 435-444
MSC (2010): Primary 22-XX
DOI: https://doi.org/10.1090/proc12739
Published electronically: May 28, 2015
MathSciNet review: 3415609
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Abstract: Let $ F$ be a non-Archimedean local field of characteristic 0 and $ G=\operatorname {SL}(n,F)$. Let $ (\pi ,W)$ be an irreducible smooth self-dual representation $ G$. The space $ W$ of $ \pi $ carries a non-degenerate $ G$-invariant bilinear form $ (\,,\,)$ which is unique up to scaling. The form $ (\,,\,)$ is easily seen to be symmetric or skew-symmetric and we set $ \varepsilon ({\pi })=\pm 1$ accordingly. In this article, we show that $ \varepsilon {(\pi )}=1$ when $ \pi $ is an Iwahori spherical representation of $ G$.


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Additional Information

Kumar Balasubramanian
Affiliation: Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal 462066, Madhya Pradesh, India
Email: bkumar@iiserb.ac.in

DOI: https://doi.org/10.1090/proc12739
Received by editor(s): March 24, 2014
Received by editor(s) in revised form: December 8, 2014
Published electronically: May 28, 2015
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society