Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter June 29, 2022

Construction of local A-packets

Hiraku Atobe EMAIL logo


In the present paper, we reformulate Mœglin’s explicit construction of local A-packets of split odd special orthogonal groups and symplectic groups. By this reformulation together with results of the previous paper with Mínguez, we can compute the A-packets explicitly. Also, we give a non-vanishing criterion of our parametrization, and an algorithm to compute certain derivatives. Finally, we prove a formula for the Aubert duals of irreducible representations of Arthur type.

Award Identifier / Grant number: 19K14494

Funding statement: The author was supported by JSPS KAKENHI Grant Number 19K14494.


We would like to thank Bin Xu for answering our questions and for suggesting his result for the proof of Theorem 6.2. We are grateful to Alberto Mínguez for useful discussions. Thanks are also due to the referee for the careful readings and the helpful comments. A Sage code for computing examples of several objects in this paper is now available at We are grateful to Alexander Hazeltine, Baiying Liu and Chi-Heng Lo for pointing out many bugs.


[1] J. Arthur, The endoscopic classification of representations. Orthogonal and symplectic groups, Amer. Math. Soc. Colloq. Publ. 61, American Mathematical Society, Providence 2013. Search in Google Scholar

[2] H. Atobe and A. Mínguez, The explicit Zelevinsky–Aubert duality, preprint (2020), Search in Google Scholar

[3] A.-M. 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; erratum, Trans. Amer. Math. Soc. 348, (1996), 4687–4690. Search in Google Scholar

[4] I. Badulescu, E. Lapid and A. Mínguez, Une condition suffisante pour l’irréductibilité d’une induite parabolique de GL ( m , D ) , Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2239–2266. 10.5802/aif.2828Search in Google Scholar

[5] C. Jantzen, Jacquet modules of p-adic general linear groups, Represent. Theory 11 (2007), 45–83. 10.1090/S1088-4165-07-00316-0Search in Google Scholar

[6] C. Jantzen, Tempered representations for classical p-adic groups, Manuscripta Math. 145 (2014), no. 3–4, 319–387. 10.1007/s00229-014-0679-5Search in Google Scholar

[7] C. Jantzen, Duality for classical p-adic groups: The half-integral case, Represent. Theory 22 (2018), 160–201. 10.1090/ert/519Search in Google Scholar

[8] T. Konno, A note on the Langlands classification and irreducibility of induced representations of p-adic groups, Kyushu J. Math. 57 (2003), no. 2, 383–409. 10.2206/kyushujm.57.383Search in Google Scholar

[9] E. Lapid and A. Mínguez, On parabolic induction on inner forms of the general linear group over a non-archimedean local field, Selecta Math. (N. S.) 22 (2016), no. 4, 2347–2400. 10.1007/s00029-016-0281-7Search in Google Scholar

[10] A. Mínguez, Sur l’irréductibilité d’une induite parabolique, J. reine angew. Math. 629 (2009), 107–131. 10.1515/CRELLE.2009.028Search in Google Scholar

[11] C. Mœglin, Paquets d’Arthur pour les groupes classiques; point de vue combinatoire, preprint (2006), Search in Google Scholar

[12] C. Mœglin, Sur certains paquets d’Arthur et involution d’Aubert–Schneider–Stuhler généralisée, Represent. Theory 10 (2006), 86–129. 10.1090/S1088-4165-06-00270-6Search in Google Scholar

[13] C. Mœglin, Paquets d’Arthur discrets pour un groupe classique p-adique, Automorphic forms and L-functions II. Local aspects, Contemp. Math. 489, American Mathematical Society, Providence (2009), 179–257. 10.1090/conm/489/09549Search in Google Scholar

[14] C. Mœglin, Holomorphie des opérateurs d’entrelacement normalisés à l’aide des paramètres d’Arthur, Canad. J. Math. 62 (2010), no. 6, 1340–1386. 10.4153/CJM-2010-074-6Search in Google Scholar

[15] C. Mœglin, Multiplicité 1 dans les paquets d’Arthur aux places p-adiques, On certain L-functions, Clay Math. Proc. 13, American Mathematical Society, Providence (2011), 333–374. Search in Google Scholar

[16] M. Tadić, Irreducibility criterion for representations induced by essentially unitary ones (case of non-Archimedean GL ( n , 𝒜 ) ), Glas. Mat. Ser. III 49(69) (2014), no. 1, 123–161. 10.3336/gm.49.1.11Search in Google Scholar

[17] M. Tadić, On unitarizability and Arthur packets, preprint (2020), 10.1007/s00229-021-01330-6Search in Google Scholar

[18] B. Xu, On Mœglin’s parametrization of Arthur packets for p-adic quasisplit Sp ( N ) and SO ( N ) , Canad. J. Math. 69 (2017), no. 4, 890–960. 10.4153/CJM-2016-029-3Search in Google Scholar

[19] B. Xu, On the cuspidal support of discrete series for p-adic quasisplit Sp ( N ) and SO ( N ) , Manuscripta Math. 154 (2017), no. 3–4, 441–502. 10.1007/s00229-017-0923-xSearch in Google Scholar

[20] B. Xu, A combinatorial solution to Mœglin’s parametrization of Arthur packets for p-adic quasisplit Sp ( N ) and O ( N ) , J. Inst. Math. Jussieu 20 (2021), no. 4, 1091–1204. 10.1017/S1474748019000409Search in Google Scholar

[21] B. Xu, Nonarchimedean components of non-endoscopic automorphic representations for quasisplit Sp ( N ) and O ( N ) , Math. Z. 297 (2021), no. 1–2, 885–921. 10.1007/s00209-020-02539-zSearch in Google Scholar

[22] A. V. Zelevinsky, Induced representations of reductive 𝔭 -adic groups. II. On irreducible representations of GL ( n ) , Ann. Sci. Éc. Norm. Supér. (4) 13 (1980), no. 2, 165–210. 10.24033/asens.1379Search in Google Scholar

Received: 2021-04-19
Revised: 2022-04-06
Published Online: 2022-06-29
Published in Print: 2022-09-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 31.1.2023 from
Scroll Up Arrow