Abstract
The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks how to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.
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author = {Henniart, Guy},
title = {Une preuve simple des conjectures de {L}anglands pour {${\rm GL}(n)$} sur un corps {$p$}-adique},
journal = {Invent. Math.},
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volume = {139},
year = {2000},
number = {2},
pages = {439--455},
issn = {0020-9910},
mrclass = {11F70 (11R39 11S37 22E50 22E55)},
mrnumber = {1738446},
mrreviewer = {Dihua Jiang},
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title = {Wave front sets of representations of {L}ie groups},
booktitle = {Automorphic Forms, Representation Theory and Arithmetic},
venue = {{B}ombay, 1979},
series = {Tata Inst. Fund. Res. Studies in Math.},
volume = {10},
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publisher = {Tata Inst. Fundamental Res., Bombay},
year = {1981},
mrclass = {22E30 (22E45 35S05 58G15)},
mrnumber = {0633659},
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zblnumber = {0494.22010},
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url = {https://doi.org/10.1007/978-3-662-00734-1_3},
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title = {Automorphic forms of low rank},
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author = {Jiang, Dihua},
title = {Automorphic integral transforms for classical groups {I}: {E}ndoscopy correspondences},
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series = {Contemp. Math.},
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number = {4},
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title = {Fourier coefficients for automorphic forms on quasisplit classical groups},
booktitle = {Advances in the Theory of Automorphic Forms and their {$L$}-Functions},
series = {Contemp. Math.},
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title = {Raising nilpotent orbits in wave-front sets},
journal = {Represent. Theory},
fjournal = {Representation Theory. An Electronic Journal of the Amer. Math. Soc.},
volume = {20},
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pages = {419--450},
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journal = {Pacific J. Math.},
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C. Moeglin and J. -L. Waldspurger, "Modèles de Whittaker dégénérés pour des groupes $p$-adiques," Math. Z., vol. 196, iss. 3, pp. 427-452, 1987.
@ARTICLE{MW87,
author = {M{\oe}glin, C. and Waldspurger, J.-L.},
title = {Modèles de {W}hittaker dégénérés pour des groupes {$p$}-adiques},
journal = {Math. Z.},
fjournal = {Mathematische Zeitschrift},
volume = {196},
year = {1987},
number = {3},
pages = {427--452},
issn = {0025-5874},
mrclass = {22E50 (11F70 11S37)},
mrnumber = {0913667},
mrreviewer = {Mitsuo Sugiura},
doi = {10.1007/BF01200363},
url = {https://doi.org/10.1007/BF01200363},
zblnumber = {0612.22008},
} -
[MW89]
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@ARTICLE{MW89,
author = {M{\oe}glin, C. and Waldspurger, J.-L.},
title = {Le spectre résiduel de {${\rm GL}(n)$}},
journal = {Ann. Sci. \'{E}cole Norm. Sup. (4)},
fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Quatrième Série},
volume = {22},
year = {1989},
number = {4},
pages = {605--674},
issn = {0012-9593},
mrclass = {22E55 (11F70 22E50)},
mrnumber = {1026752},
mrreviewer = {Stephen Gelbart},
doi = {10.24033/asens.1595},
url = {https://doi.org/10.24033/asens.1595},
zblnumber = {0696.10023},
} -
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C. Moeglin and J. -L. Waldspurger, Spectral Decomposition and Eisenstein Series, Cambridge Univ. Press, Cambridge, 1995, vol. 113.
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author = {M{\oe}glin, C. and Waldspurger, J.-L.},
title = {Spectral Decomposition and {E}isenstein Series},
series = {Cambridge Tracts in Math.},
volume = {113},
note = {Une paraphrase de l'\'{E}criture [A paraphrase of Scripture]},
publisher = {Cambridge Univ. Press, Cambridge},
year = {1995},
pages = {xxviii+338},
isbn = {0-521-41893-3},
mrclass = {11F70 (22E55)},
mrnumber = {1361168},
mrreviewer = {Joe Repka},
doi = {10.1017/CBO9780511470905},
url = {https://doi.org/10.1017/CBO9780511470905},
zblnumber = {0846.11032},
} -
[MW12] C. Moeglin and J. Waldspurger, "La conjecture locale de Gross-Prasad pour les groupes spéciaux orthogonaux: le cas général," in Sur les conjectures de Gross et Prasad. II, Math. Soc. France, Paris, 2012, vol. 347, pp. 167-216.
@INCOLLECTION{MW12,
author = {M{\oe}glin, C. and Waldspurger, Jean-Loup},
title = {La conjecture locale de {G}ross-{P}rasad pour les groupes spéciaux orthogonaux: le cas général},
booktitle = {Sur les conjectures de Gross et Prasad. II},
series = {Astérisque},
publisher = {Math. Soc. France, Paris},
volume = {347},
year = {2012},
pages = {167--216},
issn = {0303-1179},
isbn = {978-2-85629-350-8},
mrclass = {22E50},
mrnumber = {3155346},
zblnumber = {1276.22007},
} -
[MW-Book1] C. Moeglin and J. Waldspurger, Stabilisation de la Formule des Traces tordue. Vol. 1, Birkhäuser/Springer, Cham, 2016, vol. 316.
@BOOK{MW-Book1,
author = {M{\oe}glin, C. and Waldspurger, Jean-Loup},
title = {Stabilisation de la Formule des Traces tordue. {V}ol. 1},
series = {Progr. in Math.},
volume = {316},
publisher = {Birkhäuser/Springer, Cham},
year = {2016},
pages = {xviii+587},
isbn = {978-3-319-30048-1; 978-3-319-30049-8},
mrclass = {22E50 (11F70 11F72 22E55)},
mrnumber = {3823813},
mrreviewer = {Jean-Pierre Labesse},
zblnumber = {1361.11004},
} -
[MW-Book2] C. Moeglin and J. Waldspurger, Stabilisation de la Formule des Traces Tordue. Vol. 2, Birkhäuser/Springer, Cham, 2016, vol. 317.
@BOOK{MW-Book2,
author = {M{\oe}glin, C. and Waldspurger, Jean-Loup},
title = {Stabilisation de la Formule des Traces Tordue. {V}ol. 2},
series = {Progr. Math.},
volume = {317},
publisher = {Birkhäuser/Springer, Cham},
year = {2016},
pages = {v--xxviii and 589--1315},
isbn = {978-3-319-30057-3; 978-3-319-30058-0},
mrclass = {22E50 (11F70 11F72 22E55)},
mrnumber = {3823814},
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zblnumber = {1381.11001},
} -
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title = {Endoscopic classification of representations of quasi-split unitary groups},
journal = {Mem. Amer. Math. Soc.},
fjournal = {Memoirs of the Amer. Math. Soc.},
volume = {235},
year = {2015},
number = {1108},
pages = {vi+248},
issn = {0065-9266},
isbn = {978-1-4704-1041-4; 978-1-4704-2226-4},
mrclass = {22E55 (11R42 22E50)},
mrnumber = {3338302},
mrreviewer = {Neven Grbac},
doi = {10.1090/memo/1108},
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number = {3},
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series = {Amer. Math. Soc. Colloq. Publ.},
volume = {58},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2010},
pages = {vi+210},
isbn = {978-0-8218-4989-7},
mrclass = {11F70 (11-02 11M36 11R42 22E50 22E55)},
mrnumber = {2683009},
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author = {Soudry, David},
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volume = {105},
year = {1993},
number = {500},
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author = {Soudry, David},
title = {Rankin-{S}elberg integrals, the descent method, and {L}anglands functoriality},
booktitle = {International {C}ongress of {M}athematicians. {V}ol. {II}},
pages = {1311--1325},
publisher = {Eur. Math. Soc., Zürich},
year = {2006},
mrclass = {11F70 (11R39 22E55)},
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author = {Jiang, Dihua and Soudry, David and Zhang, Lei},
title={The unramified computation of {R}ankin-{S}elberg integrals for quasisplit classical groups: {B}essel model case},
note={in preparation},
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title = {Classification of unitary representations in irreducible representations of general linear group (non-{A}rchimedean case)},
journal = {Ann. Sci. \'{E}cole Norm. Sup. (4)},
fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Quatrième Série},
volume = {19},
year = {1986},
number = {3},
pages = {335--382},
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journal = {Invent. Math.},
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year = {1986},
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title = {The local {L}anglands conjecture},
booktitle = {Representation Theory of Groups and Algebras},
series = {Contemp. Math.},
volume = {145},
pages = {305--379},
publisher = {Amer. Math. Soc., Providence, RI},
year = {1993},
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author = {Waldspurger, J.-L.},
title = {La formule de {P}lancherel pour les groupes {$p$}-adiques (d'après {H}arish-{C}handra)},
journal = {J. Inst. Math. Jussieu},
fjournal = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu},
volume = {2},
year = {2003},
number = {2},
pages = {235--333},
issn = {1474-7480},
mrclass = {22E35 (22E50)},
mrnumber = {1989693},
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doi = {10.1017/S1474748003000082},
url = {https://doi.org/10.1017/S1474748003000082},
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[W] J. Waldspurger, "La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes spéciaux orthogonaux," in Sur les Conjectures de Gross et Prasad. II, Math. Soc. France, Paris, 2012, vol. 347, pp. 103-165.
@INCOLLECTION{W,
author = {Waldspurger, Jean-Loup},
title = {La conjecture locale de {G}ross-{P}rasad pour les représentations tempérées des groupes spéciaux orthogonaux},
booktitle = {Sur les Conjectures de Gross et Prasad. II},
series = {Astérisque},
publisher = {Math. Soc. France, Paris},
volume = {347},
year = {2012},
pages = {103--165},
issn = {0303-1179},
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mrclass = {22E50},
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B. Xu, Endoscopic Classification of Representations of GSp(2n) and GSO(2n), ProQuest LLC, Ann Arbor, MI, 2014.
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title = {Endoscopic {C}lassification of {R}epresentations of {GS}p(2n) and {GSO}(2n)},
note = {Thesis (Ph.D.)--Univ. of Toronto (Canada)},
publisher = {ProQuest LLC, Ann Arbor, MI},
year = {2014},
pages = {169},
isbn = {978-1321-67483-5},
mrclass = {Thesis},
mrnumber = {3347095},
url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:3688703},
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title = {The {G}an-{G}ross-{P}rasad conjecture for {${\rm U}(n)\times{\rm U}(n)$}},
journal = {Adv. Math.},
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volume = {262},
year = {2014},
pages = {1130--1191},
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doi = {10.1016/j.aim.2014.06.010},
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title = {Refined global {G}an-{G}ross-{P}rasad conjecture for {F}ourier-{J}acobi periods on symplectic groups},
journal = {Compos. Math.},
fjournal = {Compositio Mathematica},
volume = {153},
year = {2017},
number = {1},
pages = {68--131},
issn = {0010-437X},
mrclass = {11F70 (22E55)},
mrnumber = {3622873},
mrreviewer = {Nadya Gurevich},
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title = {Whittaker-{S}hintani functions for general linear groups over {$p$}-adic fields},
journal = {Sci. China Math.},
fjournal = {Science China. Mathematics},
volume = {61},
year = {2018},
number = {1},
pages = {95--110},
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mrclass = {22E50 (11F70 11F85 22E35)},
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doi = {10.1007/s11425-016-9064-6},
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W. Zhang, "Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups," Ann. of Math. (2), vol. 180, iss. 3, pp. 971-1049, 2014.
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author = {Zhang, Wei},
title = {Fourier transform and the global {G}an-{G}ross-{P}rasad conjecture for unitary groups},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {180},
year = {2014},
number = {3},
pages = {971--1049},
issn = {0003-486X},
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doi = {10.4007/annals.2014.180.3.4},
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W. Zhang, "Automorphic period and the central value of Rankin-Selberg L-function," J. Amer. Math. Soc., vol. 27, iss. 2, pp. 541-612, 2014.
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title = {Automorphic period and the central value of {R}ankin-{S}elberg {L}-function},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the Amer. Math. Soc.},
volume = {27},
year = {2014},
number = {2},
pages = {541--612},
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mrclass = {11F67 (11F70 11G40 22E55)},
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doi = {10.1090/S0894-0347-2014-00784-0},
url = {https://doi.org/10.1090/S0894-0347-2014-00784-0},
zblnumber = {1294.11069},
}