Rigid inner forms of real and $p$-adic groups

Tasho Kaletha

doi:https://doi.org/10.4007/annals.2016.184.2.6

Abstract

We define a new cohomology set $H^1(u \to W,Z \to G)$ for an affine algebraic group $G$ and a finite central subgroup $Z$, both defined over a local field of characteristic zero, which is an enlargement of the usual first Galois cohomology set of $G$. We show how this set can be used to give a precise conjectural description of the internal structure and endoscopic transfer of tempered $L$-packets for arbitrary connected reductive groups that extends the well-known conjectural description for quasi-split groups. In the case of real groups, we show that this description is correct using Shelstad’s work.

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@article{Sh90, mrkey = {1070599}, author = {Shahidi, Freydoon}, title = {A proof of {L}anglands' conjecture on {P}lancherel measures; complementary series for {$p$}-acid groups}, journal = {Ann. of Math.}, fjournal = {Annals of Mathematics. Second Series}, volume = {132}, year = {1990}, number = {2}, pages = {273--330}, issn = {0003-486X}, coden = {ANMAAH}, mrclass = {11R39 (11F70 11S37 22E35 22E55)}, mrnumber = {1070599}, mrreviewer = {Stephen Gelbart}, doi = {10.2307/1971524}, zblnumber = {0780.22005}, }
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@article{She79a, 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}, }
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@article{She79b, 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}, }
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@incollection{She79c, mrkey = {0546618}, author = {Shelstad, Diana}, title = {Notes on {$L$}-indistinguishability (based on a lecture of {R}. {P}. {L}anglands)}, booktitle = {Automorphic Forms, Representations and {$L$}-Functions}, venue = {{P}roc. {S}ympos. {P}ure {M}ath., {O}regon {S}tate {U}niv., {C}orvallis, {O}re., 1977), {P}art 2}, series = {Proc. Sympos. Pure Math.}, volume = {XXXIII}, pages = {193--203}, publisher = {Amer. Math. Soc.}, address = {Providence, R.I.}, year = {1979}, mrclass = {12A67 (10D40 22E50)}, mrnumber = {0546618}, mrreviewer = {Joe Repka}, zblnumber = {0421.12015}, }
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@article{She81, mrkey = {0627641}, author = {Shelstad, Diana}, title = {Embeddings of {$L$}-groups}, journal = {Canad. J. Math.}, fjournal = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques}, volume = {33}, year = {1981}, number = {3}, pages = {513--558}, 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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@article{She82, 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}, }
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@incollection{SheTE1, 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.}, volume = {472}, pages = {215--246}, publisher = {Amer. Math. Soc.}, address = {Providence, RI}, year = {2008}, mrclass = {22E45 (11S37)}, mrnumber = {2454336}, mrreviewer = {Ravi Raghunathan}, doi = {10.1090/conm/472/09241}, zblnumber = {1157.22008}, }
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@incollection{SheTE2, 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}, address = {Somerville, MA}, year = {2010}, mrclass = {22E45 (22E50)}, mrnumber = {2581952}, zblnumber = {1198.22008}, }
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@article{SheTE3, 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}, zblnumber = {1159.22007}, }
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@incollection{Tat63, mrkey = {0175892}, author = {Tate, John}, title = {Duality theorems in {G}alois cohomology over number fields}, booktitle = {Proc. {I}nternat. {C}ongr. {M}athematicians}, venue = {{S}tockholm, 1962}, pages = {288--295}, publisher = {Inst. Mittag-Leffler}, address = {Djursholm}, year = {1963}, mrclass = {10.66}, mrnumber = {0175892}, zblnumber = {0126.07002}, }
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@article{Tat66, mrkey = {0207680}, author = {Tate, John}, title = {The cohomology groups of tori in finite {G}alois extensions of number fields}, journal = {Nagoya Math. J.}, fjournal = {Nagoya Mathematical Journal}, volume = {27}, year = {1966}, pages = {709--719}, issn = {0027-7630}, mrclass = {12.40 (10.65)}, mrnumber = {0207680}, mrreviewer = {H. Bass}, url = {http://projecteuclid.org/euclid.nmj/1118801784}, zblnumber = {0146.06501}, }
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@incollection{Tat79, mrkey = {0546607}, author = {Tate, John}, title = {Number theoretic background}, booktitle = {Automorphic Forms, Representations and {$L$}-Functions}, venue = {{P}roc. {S}ympos. {P}ure {M}ath., {O}regon {S}tate {U}niv., {C}orvallis, {O}re., 1977, {P}art 2}, series = {Proc. Sympos. Pure Math.}, volume = {XXXIII}, pages = {3--26}, publisher = {Amer. Math. Soc.}, address = {Providence, R.I.}, year = {1979}, mrclass = {12A67}, mrnumber = {0546607}, mrreviewer = {A. I. Vinogradov}, zblnumber = {0422.12007}, }
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@article{Vog78, mrkey = {0506503}, author = {Vogan, Jr., David A.}, title = {Gel\cprime fand-{K}irillov dimension for {H}arish-{C}handra modules}, journal = {Invent. Math.}, fjournal = {Inventiones Mathematicae}, volume = {48}, year = {1978}, number = {1}, pages = {75--98}, issn = {0020-9910}, mrclass = {17B35}, mrnumber = {0506503}, mrreviewer = {Anthony Joseph}, doi = {10.1007/BF01390063}, zblnumber = {0389.17002}, }
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@article{Vog79, mrkey = {0519352}, author = {Vogan, Jr., David A.}, title = {The algebraic structure of the representation of semisimple {L}ie groups. {I}}, journal = {Ann. of Math.}, fjournal = {Annals of Mathematics. Second Series}, volume = {109}, year = {1979}, number = {1}, pages = {1--60}, issn = {0003-486X}, coden = {ANMAAH}, mrclass = {22E46 (22E47)}, mrnumber = {0519352}, mrreviewer = {A. U. Klimyk}, doi = {10.2307/1971266}, zblnumber = {0424.22010}, }
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@incollection{Vog93, mrkey = {1216197}, author = {Vogan, Jr., David A.}, 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.}, address = {Providence, RI}, year = {1993}, mrclass = {22E50 (11S37)}, mrnumber = {1216197}, mrreviewer = {Ernst-Wilhelm Zink}, doi = {10.1090/conm/145/1216197}, zblnumber = {0802.22005}, }

Rigid inner forms of real and $p$-adic groups

Abstract

Authors