# CONFLUENTES MATHEMATICI

Unitary representations of p-adic $U\left(5\right)$
Confluentes Mathematici, Tome 12 (2020) no. 1, pp. 93-146.

We study the parabolically induced complex representations of the unitary group in 5 variables$,U\left(5\right)$, defined over a $p$-adic field.

Let $F$ be a $p$-adic field. Let $E:F$ be a field extension of degree two. $U\left(5\right)$ has three proper standard Levi subgroups, the minimal Levi subgroup ${M}_{0}\cong {E}^{*}×{E}^{*}×{E}^{1}$ and the two maximal Levi subgroups ${M}_{1}\cong \mathrm{GL}\left(2,E\right)×{E}^{1}$ and ${M}_{2}\cong {E}^{*}×U\left(3\right)$.

We consider representations induced from ${M}_{0}$, representations induced from non-cuspidal, not fully-induced representations of ${M}_{1}$ and ${M}_{2}$ and representations induced from cuspidal representations of ${M}_{1}.$

We determine the points and lines of reducibility and the irreducible subquotients of these representations. Further we describe - except several particular cases - the unitary dual in terms of Langlands quotients.

Reçu le : 2017-10-30
Accepté le : 2020-06-07
Accepté après révision le : 2020-06-29
Publié le : 2020-09-25
DOI : https://doi.org/10.5802/cml.63
Classification : 22E50,  11S85,  20G05,  20C99
Mots clés: Representations, unitary group, unitary, $U\left(5\right)$, $p$-adic groups
@article{CML_2020__12_1_93_0,
author = {Claudia Schoemann},
title = {Unitary representations of p-adic $U(5)$},
journal = {Confluentes Mathematici},
pages = {93--146},
publisher = {Institut Camille Jordan},
volume = {12},
number = {1},
year = {2020},
doi = {10.5802/cml.63},
language = {en},
url = {cml.centre-mersenne.org/item/CML_2020__12_1_93_0/}
}
Claudia Schoemann. Unitary representations of p-adic $U(5)$. Confluentes Mathematici, Tome 12 (2020) no. 1, pp. 93-146. doi : 10.5802/cml.63. https://cml.centre-mersenne.org/item/CML_2020__12_1_93_0/

[1] 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., Volume 347 (1995) no. 6, pp. 2179-2189 | Article | MR 1285969 | Zbl 0827.22005

[2] W. Casselman A new nonunitarity argument for $p$-adic representations, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 28 (1981) no. 3, p. 907-928 (1982) | MR 656064

[3] W. Casselman Introduction to the theory of admissible representations of p-adic reductive groups, Draft (1995)

[4] David Goldberg Some results on reducibility for unitary groups and local Asai $L$-functions, J. Reine Angew. Math., Volume 448 (1994), pp. 65-95 | Article | MR 1266747 | Zbl 0815.11029

[5] David Goldberg R-Groups and Elliptic Representations for Unitary Groups, Proceedings of the American Mathematical Society, Volume 123 (1995) no. 4, pp. pp. 1267-1276 http://www.jstor.org/stable/2160730 | MR 1224616 | Zbl 0856.22023

[6] Marcela Hanzer The unitarizability of the Aubert dual of strongly positive square integrable representations, Israel J. Math., Volume 169 (2009), pp. 251-294 | Article | MR 2460906 | Zbl 1178.22018

[7] Marcela Hanzer; Marko Tadić A method of proving non-unitarity of representations of $p$-adic groups I, Math. Z., Volume 265 (2010) no. 4, pp. 799-816 | Article | MR 2652536 | Zbl 1215.22009

[8] A.V. Zelevinsky I.N. Bernstein Induced representations of reductive $p$-adic groups. I, Annales scientifiques de l’Ãcole Normale SupÃ©rieure, Volume 10 (1977) no. 4, pp. 441-472 http://eudml.org/doc/82002 | Numdam | MR 579172

[9] Charles David Keys On the decomposition of reducible principal series representations of $p$-adic Chevalley groups, Pacific J. Math., Volume 101 (1982) no. 2, pp. 351-388 http://projecteuclid.org/euclid.pjm/1102724780 | Article | MR 675406 | Zbl 0438.22010

[10] Charles David Keys Principal series of special unitary groups over local fields, Compositio Math., Volume 51 (1984), pp. 115-130 | Numdam | MR 734788 | Zbl 0547.22009

[11] Kazuko Konno Induced representations of U(2,2) over a p-adic field, J. Reine Angew. Math., Volume 540 (2001), pp. 167-204 | MR 1868602 | Zbl 0982.22011

[12] Ivan MatiÄ Composition series of the induced representations of $\mathrm{SO}\left(5\right)$ using intertwining operators, Glasnik Mathematicki, Volume 45/1 (2010), pp. 93-107

[13] Ivan MatiÄ The unitary dual of p-adic $\mathrm{U}\left(5\right)$, Proceedings of the American Mathematical Society, Volume 138/2 (2010), pp. 759-767 | MR 2557193

[14] Dragan Miličić On ${C}^{*}$-algebras with bounded trace, Glasnik Mat. Ser. III, Volume 8(28) (1973), pp. 7-22 | MR 0324429 | Zbl 0265.46072

[15] Goran Muić The unitary dual of $p$-adic $G_2$, Duke Math. J., Volume 90 (1997) no. 3, pp. 465-493 | Article | MR 1480543 | Zbl 0896.22006

[16] Jonathan D. Rogawski Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, Volume 123, Princeton University Press, Princeton, NJ, 1990, xii+259 pages | MR 1081540 | Zbl 0724.11031

[17] Marko Tadić Geometry of dual spaces of reductive groups (non-Archimedean case), J. Analyse Math., Volume 51 (1988), pp. 139-181 | Article | MR 963153 | Zbl 0664.22010

[18] Marko Tadić On reducibility of parabolic induction, Israel J. Math., Volume 107 (1998), pp. 29-91 | Article | MR 1658535 | Zbl 0914.22019

[19] Marko Tadić On reducibility and unitarizability for classical $p$-adic groups, some general results, Canad. J. Math., Volume 61 (2009) no. 2, pp. 427-450 | Article | MR 2504024 | Zbl 1162.22016