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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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  • [1] Kumar Balasubramanian, Self-dual representations with vectors fixed under an Iwahori subgroup, J. Algebra 394 (2013), 207-220. MR 3092718, https://doi.org/10.1016/j.jalgebra.2013.07.024
  • [2] Armand Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233-259. MR 0444849 (56 #3196)
  • [3] Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1995. Translated from the German manuscript; Corrected reprint of the 1985 translation. MR 1410059 (97i:22005)
  • [4] W. Casselman, The unramified principal series of $ {\mathfrak{p}}$-adic groups. I. The spherical function, Compositio Math. 40 (1980), no. 3, 387-406. MR 571057 (83a:22018)
  • [5] R. Gow, Real representations of the finite orthogonal and symplectic groups of odd characteristic, J. Algebra 96 (1985), no. 1, 249-274. MR 808851 (87b:20015), https://doi.org/10.1016/0021-8693(85)90049-3
  • [6] Hervé Jacquet, A correction to Conducteur des représentations du groupe linéaire [MR620708], Pacific J. Math. 260 (2012), no. 2, 515-525. MR 3001803, https://doi.org/10.2140/pjm.2012.260.515
  • [7] Hervé Jacquet, Ilja Piatetski-Shapiro, and Joseph Shalika, Conducteur des représentations génériques du groupe linéaire, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 13, 611-616 (French, with English summary). MR 625357 (82f:22015)
  • [8] Dipendra Prasad, On the self-dual representations of finite groups of Lie type, J. Algebra 210 (1998), no. 1, 298-310. MR 1656426 (2000a:20025), https://doi.org/10.1006/jabr.1998.7550
  • [9] Dipendra Prasad, On the self-dual representations of a $ p$-adic group, Internat. Math. Res. Notices 8 (1999), 443-452. MR 1687319 (2000d:22019), https://doi.org/10.1155/S1073792899000227
  • [10] Dipendra Prasad and Dinakar Ramakrishnan, Self-dual representations of division algebras and Weil groups: a contrast, Amer. J. Math. 134 (2012), no. 3, 749-767. MR 2931222, https://doi.org/10.1353/ajm.2012.0017
  • [11] Alan Roche and Steven Spallone, Twisted signs for $ p$-adic linear groups, preprint.
  • [12] F. Rodier, Modèle de Whittaker et caractères de représentations, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), Springer, Berlin, 1975, pp. 151-171. Lecture Notes in Math., Vol. 466 (French). MR 0393355 (52 #14165)
  • [13] Marko Tadić, Notes on representations of non-Archimedean $ {\rm SL}(n)$, Pacific J. Math. 152 (1992), no. 2, 375-396. MR 1141803 (92k:22029)

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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

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