Abstract
We prove the existence of a transfer of orbital integrals in endoscopy for real reductive groups when there is twisting by an automorphism defined over the reals and by a character on the real points of the group. Our proof contains a relatively short self-contained argument for the already known case of standard endoscopy.
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@misc{book,
author = {Arthur, James},
TITLE = {The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups},
NOTE={to appear},
URL = {www.claymath.org/cw/arthur/pdf/Book.pdf},
YEAR={2011},
} -
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@incollection {prob, MRKEY = {2478455},
AUTHOR = {Arthur, James},
TITLE = {Problems for real groups},
BOOKTITLE = {Representation Theory of Real Reductive {L}ie Groups},
SERIES = {Contemp. Math.},
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ZBLNUMBER = {1159.22003},
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VENUE={{P}roc. {S}ympos. {P}ure {M}ath., {O}regon {S}tate {U}niv., {C}orvallis, {O}re., 1977, {P}art 2},
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AUTHOR = {Bouaziz, Abderrazak},
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@book {aster, MRKEY = {1687096},
AUTHOR = {Kottwitz, Robert E. and Shelstad, Diana},
TITLE = {Foundations of Twisted Endoscopy},
SERIES = {Astérisque},
FJOURNAL = {Astérisque},
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YEAR = {1999},
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ISSN = {0303-1179},
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MRNUMBER = {1687096},
ZBLNUMBER = {0958.22013},
MRREVIEWER = {Volker J. Heiermann},
} -
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@misc{KS12,
author={Kottwitz, Robert E. and Shelstad, Diana},
TITLE = {On splitting invariants and sign conventions in endoscopic transfer},
YEAR={2012},
ARXIV={1201.5658},
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ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
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} -
[trans]
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@article {trans, MRKEY = {0909227},
AUTHOR = {Langlands, R. and Shelstad, D.},
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ZBLNUMBER = {0644.22005},
} -
[LS2] R. Langlands and D. Shelstad, "Descent for transfer factors," in The Grothendieck Festschrift, Vol. II, Boston, MA: Birkhäuser, 1990, pp. 485-563.
@incollection {LS2, MRKEY = {1106907},
AUTHOR = {Langlands, R. and Shelstad, D.},
TITLE = {Descent for transfer factors},
BOOKTITLE = {The {G}rothendieck {F}estschrift, {V}ol. {II}},
SERIES = {Progr. Math.},
NUMBER = {87},
PAGES = {485--563},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1990},
MRCLASS = {22E50 (11S37)},
MRNUMBER = {1106907},
MRREVIEWER = {Ernst-Wilhelm Zink},
ZBLNUMBER = {0743.22009},
} -
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author={Mezo, P.},
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JOURNAL={Mem. Amer. Math. Soc.},
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ZBLNUMBER = {1034.22011},
} -
[geom]
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@incollection {geom, MRKEY = {2454336},
AUTHOR = {Shelstad, Diana},
TITLE = {Tempered endoscopy for real groups. {I}. {G}eometric transfer with canonical factors},
BOOKTITLE = {Representation Theory of Real Reductive {L}ie Groups},
SERIES = {Contemp. Math.},
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ADDRESS = {Providence, RI},
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MRNUMBER = {2454336},
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DOI = {10.1090/conm/472/09241},
} -
[spec] D. Shelstad, "Tempered endoscopy for real groups. II. Spectral transfer factors," in Automorphic Forms and the Langlands Program, Int. Press, Somerville, MA, 2010, vol. 9, pp. 236-276.
@incollection {spec, MRKEY = {2581952},
AUTHOR = {Shelstad, Diana},
TITLE = {Tempered endoscopy for real groups. {II}. {S}pectral transfer factors},
BOOKTITLE = {Automorphic Forms and the {L}anglands Program},
SERIES = {Adv. Lect. Math. (ALM)},
VOLUME = {9},
PAGES = {236--276},
PUBLISHER = {Int. Press, Somerville, MA},
YEAR = {2010},
MRCLASS = {22E45 (22E50)},
MRNUMBER = {2581952},
ZBLNUMBER = {1198.22008},
} -
[inv]
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@article {inv, MRKEY = {2448289},
AUTHOR = {Shelstad, Diana},
TITLE = {Tempered endoscopy for real groups. {III}. {I}nversion of transfer and {$L$}-packet structure},
JOURNAL = {Represent. Theory},
FJOURNAL = {Representation Theory. An Electronic Journal of the American Mathematical Society},
VOLUME = {12},
YEAR = {2008},
PAGES = {369--402},
ISSN = {1088-4165},
MRCLASS = {22E45 (22E50)},
MRNUMBER = {2448289},
DOI = {10.1090/S1088-4165-08-00337-3},
} -
[examples]
D. Shelstad, Examples in endoscopy for real groups (notes for BIRS summer school, August 2008).
@misc{examples,
author = {Shelstad, Diana},
TITLE = {Examples in endoscopy for real groups (notes for {BIRS} summer school, {A}ugust 2008)},
URL={http://andromeda.rutgers.edu/~shelstad},
SORTYEAR={2008},
} -
[inner]
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@article {inner, MRKEY = {0539000},
AUTHOR = {Shelstad, Diana},
TITLE = {Characters and inner forms of a quasi-split group over {${\bf R}$}},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {39},
YEAR = {1979},
NUMBER = {1},
PAGES = {11--45},
ISSN = {0010-437X},
CODEN = {CMPMAF},
MRCLASS = {22E45},
MRNUMBER = {0539000},
MRREVIEWER = {A. U. Klimyk},
URL = {http://www.numdam.org/item?id=CM_1979__39_1_11_0},
ZBLNUMBER = {0431.22011},
} -
[dijon] D. Shelstad, "Orbital integrals, endoscopic groups and $L$-indistinguishability for real groups," in Conference on Automorphic Theory, Paris: Univ. Paris VII, 1983, vol. 15, pp. 135-219.
@incollection {dijon, MRKEY = {0723184},
AUTHOR = {Shelstad, Diana},
TITLE = {Orbital integrals, endoscopic groups and {$L$}-indistinguishability for real groups},
BOOKTITLE = {Conference on Automorphic Theory},
VENUE={{D}ijon, 1981},
SERIES = {Publ. Math. Univ. Paris VII},
VOLUME = {15},
PAGES = {135--219},
PUBLISHER = {Univ. Paris VII},
ADDRESS = {Paris},
YEAR = {1983},
MRCLASS = {22E45 (11F70 11F72)},
MRNUMBER = {0723184},
MRREVIEWER = {Joe Repka},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {0529.22007},
} -
@article {emb, MRKEY = {0627641},
AUTHOR = {Shelstad, Diana},
TITLE = {Embeddings of {$L$}-groups},
JOURNAL = {Canad. J. Math.},
FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
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YEAR = {1981},
NUMBER = {3},
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ISSN = {0008-414X},
CODEN = {CJMAAB},
MRCLASS = {22E50 (10D99 20G20)},
MRNUMBER = {0627641},
MRREVIEWER = {J. Tunnell},
DOI = {10.4153/CJM-1981-044-4},
ZBLNUMBER = {0457.22006},
} -
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D. Shelstad, "$L$-indistinguishability for real groups," Math. Ann., vol. 259, iss. 3, pp. 385-430, 1982.
@article {lind, MRKEY = {0661206},
AUTHOR = {Shelstad, Diana},
TITLE = {{$L$}-indistinguishability for real groups},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {259},
YEAR = {1982},
NUMBER = {3},
PAGES = {385--430},
ISSN = {0025-5831},
CODEN = {MAANA3},
MRCLASS = {22E45},
MRNUMBER = {0661206},
MRREVIEWER = {Stephen Gelbart},
DOI = {10.1007/BF01456950},
ZBLNUMBER = {0506.22014},
} -
@misc {twspec, MRKEY = {2581952},
AUTHOR = {Shelstad, Diana},
TITLE = {On spectral transfer factors in real twisted endoscopy},
YEAR={2011},
URL={http://andromeda.rutgers.edu/~shelstad},
} -
[family]
D. Shelstad, "Orbital integrals and a family of groups attached to a real reductive group," Ann. Sci. École Norm. Sup., vol. 12, iss. 1, pp. 1-31, 1979.
@article {family, MRKEY = {0532374},
AUTHOR = {Shelstad, Diana},
TITLE = {Orbital integrals and a family of groups attached to a real reductive group},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {12},
YEAR = {1979},
NUMBER = {1},
PAGES = {1--31},
ISSN = {0012-9593},
CODEN = {ENAQAF},
MRCLASS = {22E45 (22E46)},
MRNUMBER = {0532374},
MRREVIEWER = {Wulf Rossmann},
URL = {http://www.numdam.org/item?id=ASENS_1979_4_12_1_1_0},
ZBLNUMBER = {0433.22006},
} -
[S11]
D. Shelstad, "Base change and a matching theorem for real groups," in Noncommutative Harmonic Analysis and Lie Groups, New York: Springer-Verlag, 1981, vol. 880, pp. 425-482.
@incollection {S11, MRKEY = {0644843},
AUTHOR = {Shelstad, Diana},
TITLE = {Base change and a matching theorem for real groups},
BOOKTITLE = {Noncommutative Harmonic Analysis and {L}ie Groups },
VENUE={{M}arseille, 1980},
SERIES = {Lecture Notes in Math.},
VOLUME = {880},
PAGES = {425--482},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1981},
MRCLASS = {22E45 (10D40)},
MRNUMBER = {0644843},
MRREVIEWER = {Wen Ch'ing Winnie Li},
ZBLNUMBER = {0494.22007},
DOI = {10.1007/BFb0090419},
} -
[S12]
D. Shelstad, "Endoscopic groups and base change ${\bf C}/{\bf R}$," Pacific J. Math., vol. 110, iss. 2, pp. 397-416, 1984.
@article {S12, MRKEY = {0726498},
AUTHOR = {Shelstad, Diana},
TITLE = {Endoscopic groups and base change {${\bf C}/{\bf R}$}},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {110},
YEAR = {1984},
NUMBER = {2},
PAGES = {397--416},
ISSN = {0030-8730},
CODEN = {PJMAAI},
MRCLASS = {22E45 (22E46)},
MRNUMBER = {0726498},
MRREVIEWER = {Wulf Rossmann},
URL = {http://projecteuclid.org/euclid.pjm/1102710927},
ZBLNUMBER = {0488.22033},
} -
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@book{V,
author={Varadarajan, V. S.},
TITLE={Harmonic Analysis on Reductive {L}ie Groups},
SERIES = {Lect. Notes in Math.},
VOLUME={576},
PUBLISHER = {Springer-Verlag},
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YEAR = {1977},
PAGES = {v+521},
MRCLASS = {22E45},
MRNUMBER = {0473111},
ZBLNUMBER = {0354.43001},
DOI = {10.1007/BFb0097814},
MRREVIEWER = {A. U. Klimyk},
} -
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author={Waldspurger, J-L.},
TITLE={L'endoscopie tordue n'est pas si tordue},
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YEAR = {2008},
NUMBER = {908},
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ISSN = {0065-9266},
CODEN = {MAMCAU},
MRCLASS = {22E50 (11R34 22E35)},
MRNUMBER = {2418405},
ZBLNUMBER = {1146.22016},
MRREVIEWER = {Anne-Marie H. Aubert},
}%%%% -
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YEAR={2009},
} -
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@book {Wall, MRKEY = {0929683},
AUTHOR = {Wallach, Nolan R.},
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SERIES = {Pure Appl. Math.},
VOLUME = {132},
PUBLISHER = {Academic Press},
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PAGES = {xx+412},
ISBN = {0-12-732960-9},
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@book {War, MRKEY = {0499000},
AUTHOR = {Warner, Garth},
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SERIES={Grundlehren Math. Wissen.},
VOLUME={189},
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}