The Shimura–Waldspurger correspondence for $\mathrm {Mp}_{2n}$

Abstract

We generalize the Shimura–Waldspurger correspondence, which describes the generic part of the automorphic discrete spectrum of the metaplectic group $\mathrm {Mp}_2$, to the metaplectic group $\mathrm {Mp}_{2n}$ of higher rank. To establish this, we transport Arthur’s endoscopic classification of representations of the odd special orthogonal group $\mathrm {SO}_{2r+1}$ with $r \gg 2n$ by using a result of J.-S. Li on global theta lifts in the stable range.

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      author = {Ginzburg, David and Rallis, Stephen and Soudry, David},
      title = {Endoscopic representations of {${\widetilde{\rm Sp}}_{2n}$}},
      journal = {J. Inst. Math. Jussieu},
      fjournal = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu},
      volume = {1},
      year = {2002},
      number = {1},
      pages = {77--123},
      issn = {1474-7480},
      mrclass = {11F70 (22E55)},
      mrnumber = {1954940},
      mrreviewer = {Dihua Jiang},
      doi = {10.1017/S1474748002000026},
      url = {https://doi.org/10.1017/S1474748002000026},
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      author = {Gross, Benedict H. and Prasad, Dipendra},
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      journal = {Canad. J. Math.},
      fjournal = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
      volume = {44},
      year = {1992},
      number = {5},
      pages = {974--1002},
      issn = {0008-414X},
      mrclass = {22E50 (11R39 22E55)},
      mrnumber = {1186476},
      mrreviewer = {David Joyner},
      doi = {10.4153/CJM-1992-060-8},
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    @BOOK{ht,
      author = {Harris, Michael and Taylor, Richard},
      title = {The Geometry and Cohomology of Some Simple {S}himura Varieties},
      series = {Ann. of Math. Stud.},
      volume = {151},
      note = {with an appendix by Vladimir G. Berkovich},
      publisher = {Princeton Univ. Press, Princeton, NJ},
      year = {2001},
      pages = {viii+276},
      isbn = {0-691-09090-4},
      mrclass = {11G18 (11F70 11S37 14G35 22E45)},
      mrnumber = {1876802},
      mrreviewer = {James Milne},
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      doi = {10.1515/9781400837205},
      }
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      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {139},
      year = {2000},
      number = {2},
      pages = {439--455},
      issn = {0020-9910},
      mrclass = {11F70 (11R39 11S37 22E50 22E55)},
      mrnumber = {1738446},
      mrreviewer = {Dihua Jiang},
      doi = {10.1007/s002220050012},
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      author = {Howe, R.},
      title = {{$\theta $}-series and invariant theory},
      booktitle = {Automorphic Forms, Representations and {$L$}-Functions, {P}art 1},
      venue = {{P}roc. {S}ympos. {P}ure {M}ath., {O}regon {S}tate {U}niv., {C}orvallis, {O}re., 1977},
      series = {Proc. Sympos. Pure Math.},
      volume = {XXXIII},
      pages = {275--285},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1979},
      mrclass = {22E55 (10D40)},
      mrnumber = {0546602},
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      }
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      author = {Howe, Roger},
      title = {Automorphic forms of low rank},
      booktitle = {Noncommutative Harmonic Analysis and {L}ie Groups},
      venue = {{M}arseille, 1980},
      series = {Lecture Notes in Math.},
      volume = {880},
      pages = {211--248},
      publisher = {Springer, Berlin-New York},
      year = {1981},
      mrclass = {10D40 (22E55)},
      mrnumber = {0644835},
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      }
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      author = {Howe, Roger},
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      booktitle = {Harmonic Analysis and Group Representations},
      pages = {223--331},
      publisher = {Liguori, Naples},
      year = {1982},
      mrclass = {22E45 (22E46 22E50)},
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      zblnumber = {},
      }
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      title = {Transcending classical invariant theory},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the Amer. Math. Soc.},
      volume = {2},
      year = {1989},
      number = {3},
      pages = {535--552},
      issn = {0894-0347},
      mrclass = {22E45},
      mrnumber = {0985172},
      doi = {10.2307/1990942},
      url = {https://doi.org/10.2307/1990942},
      zblnumber = {0716.22006},
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      journal = {Duke Math. J.},
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      doi = {10.1215/00127094-0000001X},
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      title = {Sur un résultat de {W}aldspurger},
      journal = {Ann. Sci. École Norm. Sup. (4)},
      fjournal = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
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      issn = {0012-9593},
      mrclass = {11F70 (22E55)},
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      url = {http://www.numdam.org/item?id=ASENS_1986_4_19_2_185_0},
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      journal = {Proc. Indian Acad. Sci. Math. Sci.},
      fjournal = {Indian Academy of Sciences. Proceedings. Mathematical Sciences},
      volume = {97},
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      pages = {117--155 (1988)},
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      mrclass = {11F72 (11F37 11F67)},
      mrnumber = {0983610},
      mrreviewer = {Daniel Bump},
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      journal = {Amer. J. Math.},
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      issn = {0002-9327},
      mrclass = {10D40 (12A67 22E55)},
      mrnumber = {0623137},
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      doi = {10.2307/2374050},
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      author = {Jacquet, Hervé and Shalika, Joseph},
      title = {Exterior square {$L$}-functions},
      booktitle = {Automorphic Forms, {S}himura Varieties, and {$L$}-Functions, {V}ol. {II}},
      venue = {{A}nn {A}rbor, {MI},
      1988},
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      volume = {11},
      pages = {143--226},
      publisher = {Academic Press, Boston, MA},
      year = {1990},
      mrclass = {11F67 (11F27 11F70 22E50)},
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      mrclass = {22E50 (11F27 11F70)},
      mrnumber = {0818351},
      mrreviewer = {Marie-France Vignéras},
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      journal = {Ann. of Math. (2)},
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      year = {1994},
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      mrclass = {11F70 (11F27 22E55)},
      mrnumber = {1289491},
      mrreviewer = {Colette M{{\oe}}glin},
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      author = {Langlands, R. P.},
      title = {Automorphic representations, {S}himura varieties, and motives. {E}in {M}ärchen},
      booktitle = {Automorphic Forms, Representations and {$L$}-Functions, {P}art 2},
      venue = {{P}roc. {S}ympos. {P}ure {M}ath., {O}regon {S}tate {U}niv., {C}orvallis, {O}re., 1977},
      series = {Proc. Sympos. Pure Math.},
      volume = {XXXIII},
      pages = {205--246},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1979},
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      pages = {101--170},
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      year = {1989},
      mrclass = {22E45 (22E46)},
      mrnumber = {1011897},
      mrreviewer = {D. Miličić},
      doi = {10.1090/surv/031/03},
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      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {97},
      year = {1989},
      number = {2},
      pages = {237--255},
      issn = {0020-9910},
      mrclass = {22E50 (22E46)},
      mrnumber = {1001840},
      mrreviewer = {Marko Tadić},
      doi = {10.1007/BF01389041},
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      journal = {Compositio Math.},
      fjournal = {Compositio Mathematica},
      volume = {71},
      year = {1989},
      number = {1},
      pages = {29--48},
      issn = {0010-437X},
      mrclass = {22E50},
      mrnumber = {1008803},
      mrreviewer = {J. A. Wolf},
      url = {http://www.numdam.org/item?id=CM_1989__71_1_29_0},
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      year = {1997},
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      issn = {0002-9327},
      mrclass = {11F70 (11F30 11F46 22E55)},
      mrnumber = {1448215},
      mrreviewer = {Henry H. Kim},
      url = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v119/119.3li.pdf},
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      fjournal = {Compositio Mathematica},
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      year = {2011},
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      issn = {0010-437X},
      mrclass = {22E35 (11F72 22E50)},
      mrnumber = {2776612},
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      doi = {10.1112/S0010437X10004963},
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      doi = {10.1007/s00222-015-0577-9},
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      author = {M{{\oe}}glin, Colette},
      title = {Paquets d'{A}rthur discrets pour un groupe classique {$p$}-adique},
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      url = {https://doi.org/10.1090/conm/489/09549},
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      author = {M{{\oe}}glin, Colette},
      title = {Multiplicité 1 dans les paquets d'{A}rthur aux places {$p$}-adiques},
      booktitle = {On Certain {$L$}-Functions},
      series = {Clay Math. Proc.},
      volume = {13},
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      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2011},
      mrclass = {22E50 (11F70 22E35)},
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      journal = {Adv. Math.},
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Authors

Wee Teck Gan

Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076

Atsushi Ichino

Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan