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SUR LES PAQUETS D’ARTHUR DE $\mathbf{Sp}(2n,\mathbb{R})$ CONTENANT DES MODULES UNITAIRES DE PLUS HAUT POIDS, SCALAIRES

  • COLETTE MOEGLIN (a1) and DAVID RENARD (a2)

Abstract

Soit $\unicode[STIX]{x1D70B}$ un module de plus haut poids unitaire du groupe $G=\mathbf{Sp}(2n,\mathbb{R})$ . On s’intéresse aux paquets d’Arthur contenant  $\unicode[STIX]{x1D70B}$ . Lorsque le plus haut poids est scalaire, on détermine les paramètres de ces paquets, on établit la propriété de multiplicité $1$ de $\unicode[STIX]{x1D70B}$ dans le paquet, et l’on calcule le caractère $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D70B}}$ (du groupe des composantes connexes du centralisateur du paramètre dans le groupe dual) associé à $\unicode[STIX]{x1D70B}$ et qui joue un grand rôle dans la théorie d’Arthur. On fait de même pour certains modules de plus haut poids unitaires unipotents  $\unicode[STIX]{x1D70E}_{n,k}$ , ou bien lorsque le caractère infinitésimal est régulier.

Let $\unicode[STIX]{x1D70B}$ be an irreducible unitary highest weight module for $G=\mathbf{Sp}(2n,\mathbb{R})$ . We would like to determine the Arthur packets containing  $\unicode[STIX]{x1D70B}$ . When the highest weight is scalar, we determine the Arthur parameter of these packets, we establish the multiplicity one property of  $\unicode[STIX]{x1D70B}$ in the packet and we compute the character $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D70B}}$ (of the group of connected components of the centralizer of $\unicode[STIX]{x1D713}$ in the dual group) associated to $\unicode[STIX]{x1D70B}$ which plays an important role in Arthur’s theory. We also deal with the case of some unipotent unitary highest weight modules  $\unicode[STIX]{x1D70E}_{n,k}$ , or when the infinitesimal character is regular.

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Le second auteur a bénéficié d’une aide de l’Agence nationale de la recherche ANR-13-BS01-0012 FERPLAY.

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References

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[Ada87] Adams, J., Unitary highest weight modules , Adv. Math. 63(2) (1987), 113137.
[ABV92] Adams, J., Barbasch, D. and Vogan, D. A. Jr, The Langlands Classification and Irreducible Characters for Real Reductive Groups, Progress in Mathematics 104 , Birkhäuser Boston, Boston, MA, 1992.
[AJ87] Adams, J. and Johnson, J. F., Endoscopic groups and packets of nontempered representations , Compositio Math. 64(3) (1987), 271309.
[AMR] Arancibia, N., Mœglin, C. and Renard, D., Paquets d’Arthur des groupes classiques et unitaires, cas cohomologique , Annales de la faculté des sciences de Toulouse, Série 6 27(5) (2015), 10231105.
[Art13] Arthur, J., “ Orthogonal and symplectic groups ”, in The Endoscopic Classification of Representations, American Mathematical Society Colloquium Publications 61 , American Mathematical Society, Providence, RI, 2013.
[Bar89] Barbasch, D., The unitary dual for complex classical Lie groups , Invent. Math. 96(1) (1989), 103176.
[CL] Chenevier, G. and Lannes, J., Formes automorphes et voisins de Kneser des réseaux de Niemeier, Springer, Switzerland, 2019.
[CR15] Chenevier, G. and Renard, D., Level one algebraic cusp forms of classical groups of small rank , Mem. Amer. Math. Soc. 237(1121) (2015), v+122.
[EHW83] Enright, T., Howe, R. and Wallach, N., “ A classification of unitary highest weight modules ”, in Representation Theory of Reductive Groups (Park City, Utah, 1982), Progress in Mathematics 40 , Birkhäuser Boston, Boston, MA, 1983, 97143.
[GT16] Gan, W. T. and Takeda, S., A proof of the Howe duality conjecture , J. Amer. Math. Soc. 29(2) (2016), 473493.
[HS83] Hecht, H. and Schmid, W., Characters, asymptotics and n-homology of Harish-Chandra modules , Acta Math. 151(1–2) (1983), 49151.
[How89] Howe, R., Transcending classical invariant theory , J. Amer. Math. Soc. 2(3) (1989), 535552.
[Igu62] Igusa, J.-I., On Siegel modular forms of genus two , Amer. J. Math. 84 (1962), 175200.
[Jak83] Jakobsen, H. P., Hermitian symmetric spaces and their unitary highest weight modules , J. Funct. Anal. 52(3) (1983), 385412.
[KV78] Kashiwara, M. and Vergne, M., On the Segal–Shale–Weil representations and harmonic polynomials , Invent. Math. 44(1) (1978), 147.
[KV95] Knapp, A. W. and Vogan, D. A. Jr, Cohomological Induction and Unitary Representations, Princeton Mathematical Series 45 , Princeton University Press, Princeton, NJ, 1995.
[KR94] Kudla, S. S. and Rallis, S., A regularized Siegel–Weil formula: the first term identity , Ann. of Math. (2) 140(1) (1994), 180.
[Mat04] Matumoto, H., On the representations of Sp(p, q) and SO(2n) unitarily induced from derived functor modules , Composito Math. 140(4) (2004), 10591096.
[Mœg06] Mœglin, C., Sur certains paquets d’Arthur et involution d’Aubert-Schneider-Stuhler généralisée , Represent. Theory 10 (2006), 86129.
[Mœg08] Mœglin, C., Formes automorphes de carré intégrable non cuspidales , Manuscripta Math. 127(4) (2008), 411467.
[Mœg11] Mœglin, C., “ Conjecture d’Adams pour la correspondance de Howe et filtration de Kudla ”, in Arithmetic Geometry and Automorphic Forms, Adv. Lect. Math. (ALM) 19 , Int. Press, Somerville, MA, 2011, 445503.
[Mœg17] Mœglin, C., “ Paquets d’Arthur spéciaux unipotents aux places archimédiennes et correspondance de Howe ”, in Representation Theory, Number Theory, and Invariant Theory, Progress in Mathematics 323 , Birkhäuser/Springer, Cham, 2017, 469502.
[MRa] Mœglin, C. and Renard, D., “ Sur les paquets d’Arthur aux places réelles, translation ”, in Geometric Aspects of the Trace Formula, Simons Symposia, Springer.
[MRb] Mœglin, C. and Renard, D., Sur les paquets d’Arthur des groupes classiques réels , J. Eur. Math. Soc. (à paraître).
[MW95] Mœglin, C. and Waldspurger, J.-L., “ Une paraphrase de l’Écriture [A paraphrase of Scripture] ”, in Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Mathematics 113 , Cambridge University Press, Cambridge, 1995.
[NOT01] Nishiyama, K., Ochiai, H. and Taniguchi, K., Bernstein degree and associated cycles of Harish-Chandra modules—Hermitian symmetric case , Astérisque (273) (2001), 1380. Nilpotent orbits, associated cycles and Whittaker models for highest weight representations.
[Oda94] Oda, T., An explicit integral representation of Whittaker functions on Sp(2; R) for the large discrete series representations , Tohoku Math. J. (2) 46(2) (1994), 261279.
[Oht91] Ohta, T., The closures of nilpotent orbits in the classical symmetric pairs and their singularities , Tohoku Math. J. (2) 43(2) (1991), 161211.
[Ral84] Rallis, S., On the Howe duality conjecture , Compositio Math. 51(3) (1984), 333399.
[She15] Shelstad, D., “ On elliptic factors in real endoscopic transfer I ”, in Representations of Reductive Groups, Progress in Mathematics 312 , Birkhäuser/Springer, Cham, 2015, 455504.
[SZ15] Sun, B. and Zhu, C.-B., Conservation relations for local theta correspondence , J. Amer. Math. Soc. 28(4) (2015), 939983.
[Taï17] Taïbi, O., Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula , Ann. Sci. Éc. Norm. Supér. (4) 50(2) (2017), 269344.
[Taï] Taïbi, O., Arthur’s multiplicity formula for certain inner forms of special orthogonal and symplectic groups , J. Eur. Math. Soc. (JEMS) 21(3) (2019), 839871.
[Tra01] Trapa, P. E., Annihilators and associated varieties of A q(𝜆) modules for U(p, q) , Compositio Math. 129(1) (2001), 145.
[Tra05] Trapa, P. E., Richardson orbits for real classical groups , J. Algebra 286(2) (2005), 361385.
[Tsu84] Tsushima, R., “ An explicit dimension formula for the spaces of generalized automorphic forms with respect to Sp(2, Z) ”, in Automorphic Forms of Several Variables (Katata, 1983), Progress in Mathematics 46 , Birkhäuser Boston, Boston, MA, 1984, 378383.
[Tsu87] Tsuyumine, S., On the Siegel modular function field of degree three , Compositio Math. 63(1) (1987), 8398.
[Vog81] Vogan, D. A. Jr, Representations of Real Reductive Lie Groups, Progress in Mathematics 15 , Birkhäuser, Boston, MA, 1981.
[Vog88] Vogan, D. A. Jr, “ Irreducibility of discrete series representations for semisimple symmetric spaces ”, in Representations of Lie Groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math. 14 , Academic Press, Boston, MA, 1988, 191221.
[Vog91] Vogan, D. A. Jr, “ Associated varieties and unipotent representations ”, in Harmonic Analysis on Reductive Groups (Brunswick, ME, 1989), Progress in Mathematics 101 , Birkhäuser Boston, Boston, MA, 1991, 315388.
[VZ84] Vogan, D. A. Jr and Zuckerman, G. J., Unitary representations with nonzero cohomology , Compositio Math. 53(1) (1984), 5190.
[Wal90] Waldspurger, J.-L., “ Démonstration d’une conjecture de dualité de Howe dans le cas p-adique, p≠2 ”, in Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday, Part I (Ramat Aviv, 1989), Israel Math. Conf. Proc. 2 , Weizmann, Jerusalem, 1990, 267324.
[Wal03] Wallach, N. R., Generalized Whittaker vectors for holomorphic and quaternionic representations , Comment. Math. Helv. 78(2) (2003), 266307.
[Yam14] Yamana, S., L-functions and theta correspondence for classical groups , Invent. Math. 196(3) (2014), 651732.
[Zhu03] Zhu, C.-B., Representations with scalar K-types and applications , Israel J. Math. 135 (2003), 111124.
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