Aubert duals of strongly positive discrete series and a class of unitarizable representations

Matić, Ivan

doi:10.1090/proc/13461

Aubert duals of strongly positive discrete series and a class of unitarizable representations
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Proc. Amer. Math. Soc. 145 (2017), 3561-3570 Request permission

Abstract:

Let $G_n$ denote either the group $Sp(n, F)$ or $SO(2n+1, F)$ over a non-archimedean local field $F$. We explicitly determine the Aubert duals of strongly positive discrete series representations of the group $G_n$. This enables us to construct a large class of unitarizable representations of this group.

References

James Arthur, The endoscopic classification of representations, American Mathematical Society Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013. Orthogonal and symplectic groups. MR 3135650, DOI 10.1090/coll/061
Anne-Marie Aubert, Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif $p$-adique, Trans. Amer. Math. Soc. 347 (1995), no. 6, 2179–2189 (French, with English summary). MR 1285969, DOI 10.1090/S0002-9947-1995-1285969-0
Anne-Marie Aubert, Erratum: “Duality in the Grothendieck group of the category of finite-length smooth representations of a $p$-adic reductive group” [Trans. Amer. Math. Soc. 347 (1995), no. 6, 2179–2189; MR1285969 (95i:22025)], Trans. Amer. Math. Soc. 348 (1996), no. 11, 4687–4690 (French). MR 1390967, DOI 10.1090/S0002-9947-96-01776-X
Marcela Hanzer, The unitarizability of the Aubert dual of strongly positive square integrable representations, Israel J. Math. 169 (2009), 251–294. MR 2460906, DOI 10.1007/s11856-009-0011-3
Ivan Matić, Strongly positive representations of metaplectic groups, J. Algebra 334 (2011), 255–274. MR 2787663, DOI 10.1016/j.jalgebra.2011.02.015
Ivan Matić, Jacquet modules of strongly positive representations of the metaplectic group $\widetilde {Sp(n)}$, Trans. Amer. Math. Soc. 365 (2013), no. 5, 2755–2778. MR 3020114, DOI 10.1090/S0002-9947-2012-05725-4
Ivan Matić and Marko Tadić, On Jacquet modules of representations of segment type, Manuscripta Math. 147 (2015), no. 3-4, 437–476. MR 3360752, DOI 10.1007/s00229-015-0727-9
Colette Mœglin, Paquets stables des séries discrètes accessibles par endoscopie tordue; leur paramètre de Langlands, Automorphic forms and related geometry: assessing the legacy of I. I. Piatetski-Shapiro, Contemp. Math., vol. 614, Amer. Math. Soc., Providence, RI, 2014, pp. 295–336 (French, with English summary). MR 3220932, DOI 10.1090/conm/614/12254
Colette Mœglin and Marko Tadić, Construction of discrete series for classical $p$-adic groups, J. Amer. Math. Soc. 15 (2002), no. 3, 715–786. MR 1896238, DOI 10.1090/S0894-0347-02-00389-2
C. Mœglin and J.-L. Waldspurger, Sur l’involution de Zelevinski, J. Reine Angew. Math. 372 (1986), 136–177 (French). MR 863522, DOI 10.1515/crll.1986.372.136
Allan J. Silberger, Special representations of reductive $p$-adic groups are not integrable, Ann. of Math. (2) 111 (1980), no. 3, 571–587. MR 577138, DOI 10.2307/1971110