Self-dual representations of 𝑆𝐿(𝑛,𝐹)

Balasubramanian, Kumar

doi:10.1090/proc12739

Self-dual representations of $\operatorname {SL}(n,F)$
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Proc. Amer. Math. Soc. 144 (2016), 435-444 Request permission

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 $G$.

References

Kumar Balasubramanian, Self-dual representations with vectors fixed under an Iwahori subgroup, J. Algebra 394 (2013), 207–220. MR 3092718, DOI 10.1016/j.jalgebra.2013.07.024
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 444849, DOI 10.1007/BF01390139
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
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
R. Gow, Real representations of the finite orthogonal and symplectic groups of odd characteristic, J. Algebra 96 (1985), no. 1, 249–274. MR 808851, DOI 10.1016/0021-8693(85)90049-3
Hervé Jacquet, A correction to Conducteur des représentations du groupe linéaire [MR620708], Pacific J. Math. 260 (2012), no. 2, 515–525. MR 3001803, DOI 10.2140/pjm.2012.260.515
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
Dipendra Prasad, On the self-dual representations of finite groups of Lie type, J. Algebra 210 (1998), no. 1, 298–310. MR 1656426, DOI 10.1006/jabr.1998.7550
Dipendra Prasad, On the self-dual representations of a $p$-adic group, Internat. Math. Res. Notices 8 (1999), 443–452. MR 1687319, DOI 10.1155/S1073792899000227
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, DOI 10.1353/ajm.2012.0017
Alan Roche and Steven Spallone, Twisted signs for $p$-adic linear groups, preprint.
F. Rodier, Modèle de Whittaker et caractères de représentations, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), Lecture Notes in Math., Vol. 466, Springer, Berlin, 1975, pp. 151–171 (French). MR 0393355
Marko Tadić, Notes on representations of non-Archimedean $\textrm {SL}(n)$, Pacific J. Math. 152 (1992), no. 2, 375–396. MR 1141803