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Klyachko models of $ p$-adic special linear groups

Authors: Joshua M. Lansky and C. Ryan Vinroot
Journal: Proc. Amer. Math. Soc. 139 (2011), 2271-2279
MSC (2000): Primary 22E50
Published electronically: November 29, 2010
Previous version: Original version posted November 24, 2010
Current version: Adds Acknowledgments section inadvertently deleted by the publisher during processing
MathSciNet review: 2775404
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Abstract: We study Klyachko models of $ \mathrm{SL}(n, F)$, where $ F$ is a non- Archimedean local field. In particular, using results of Klyachko models for $ \mathrm{GL}(n, F)$ due to Heumos, Rallis, Offen and Sayag, we give statements of existence, uniqueness, and disjointness of Klyachko models for admissible representations of $ \mathrm{SL}(n, F)$, where the uniqueness and disjointness are up to specified conjugacy of the inducing character, and the existence is for unitarizable representations in the case $ F$ has characteristic 0. We apply these results to relate the size of an $ L$-packet containing a given representation of $ \mathrm{SL}(n, F)$ to the type of its Klyachko model, and we describe when a self-dual unitarizable representation of $ \mathrm{SL}(n, F)$ is orthogonal and when it is symplectic.

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Additional Information

Joshua M. Lansky
Affiliation: Department of Mathematics and Statistics, American University, 4400 Massachusetts Avenue, NW, Washington, DC 20016

C. Ryan Vinroot
Affiliation: Department of Mathematics, College of William and Mary, P. O. Box 8795, Williamsburg, Virginia 23187

Keywords: Klyachko model, $p$-adic special linear group, multiplicity one, $L$-packets, self-dual representations
Received by editor(s): September 3, 2009
Received by editor(s) in revised form: June 10, 2010
Published electronically: November 29, 2010
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2010 American Mathematical Society