On irreducible representations of compact $p$-adic analytic groups

Konstantin Ardakov; Simon Wadsley

doi:https://doi.org/10.4007/annals.2013.178.2.3

Abstract

We prove that the canonical dimension of a coadmissible representation of a semisimple $p$-adic Lie group in a $p$-adic Banach space is either zero or at least half the dimension of a nonzero coadjoint orbit. To do this we establish analogues for $p$-adically completed enveloping algebras of Bernstein’s inequality for modules over Weyl algebras, the Beilinson-Bernstein localisation theorem and Quillen’s Lemma about the endomorphism ring of a simple module over an enveloping algebra.

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@misc{ArdThesis, author={Ardakov, K.}, TITLE={Krull dimension of {I}wasawa algebras and some related topics}, NOTE={Ph.D. thesis, University of Cambridge}, YEAR={2004}, }
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@article {Gabber, MRKEY = {0618321}, AUTHOR = {Gabber, Ofer}, TITLE = {The integrability of the characteristic variety}, JOURNAL = {Amer. J. Math.}, FJOURNAL = {American Journal of Mathematics}, VOLUME = {103}, YEAR = {1981}, NUMBER = {3}, PAGES = {445--468}, ISSN = {0002-9327}, CODEN = {AJMAAN}, MRCLASS = {58G07 (16A03 16A05 32A45)}, MRNUMBER = {0618321}, MRREVIEWER = {Zoghman Mebkhout}, DOI = {10.2307/2374101}, ZBLNUMBER = {0492.16002}, }
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@article {Gabriel, MRKEY = {0232821}, AUTHOR = {Gabriel, Pierre}, TITLE = {Des catégories abéliennes}, JOURNAL = {Bull. Soc. Math. France}, FJOURNAL = {Bulletin de la Société Mathématique de France}, VOLUME = {90}, YEAR = {1962}, PAGES = {323--448}, ISSN = {0037-9484}, MRCLASS = {18.20}, MRNUMBER = {0232821}, MRREVIEWER = {T.-Y. Lam}, ZBLNUMBER = {0201.35602}, }
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@article {GK, MRKEY = {0241487}, AUTHOR = {Gel{\cprime}fand, I. M. and Kirillov, A. A.}, TITLE = {Structure of the {L}ie sfield connected with a semisimple decomposable {L}ie algebra}, JOURNAL = {Funkcional. Anal. i Priložen.}, FJOURNAL = {Akademija Nauk SSSR. Funkcional\cprime nyi Analiz i ego Priloženija}, VOLUME = {3}, YEAR = {1969}, NUMBER = {1}, PAGES = {7--26}, ISSN = {0374-1990}, MRCLASS = {17.30}, MRNUMBER = {0241487}, MRREVIEWER = {K. A. {Ž}evlakov}, }
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@article {Ginz, MRKEY = {0241487}, AUTHOR = {Ginsburg, V.}, TITLE = {Characteristic varieties and vanishing cycles}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {84}, YEAR = {1986}, NUMBER = {2}, PAGES = {327--402}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {32C38 (35A27 58G07)}, MRNUMBER = {0241487}, MRREVIEWER = {Daniel Barlet}, DOI = {10.1007/BF01388811}, ZBLNUMBER = {0598.32013}, }
[GreuelPfister] G. Greuel and G. Pfister, A Singular Introduction to Commutative Algebra, extended ed., New York: Springer-Verlag, 2008.

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@book {GreuelPfister, MRKEY = {2363237}, AUTHOR = {Greuel, Gert-Martin and Pfister, Gerhard}, TITLE = {A {\bf {S}ingular} Introduction to Commutative Algebra}, EDITION = {extended}, NOTE = {with contributions by Olaf Bachmann, Christoph Lossen and Hans Sch{ö}nemann, With 1 CD-ROM (Windows, Macintosh and UNIX)}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {2008}, PAGES = {xx+689}, ISBN = {978-3-540-73541-0}, MRCLASS = {13-01 (13-04)}, MRNUMBER = {2363237}, }
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@article{EGAI, MRKEY={0217083}, AUTHOR={Grothendieck, A.}, TITLE={\'{E}léments de géométrie algébrique. {I}. {L}e langage des schémas}, JOURNAL={Inst. Hautes Études Sci. Publ. Math.}, VOLUME={4}, PAGES={1--228}, YEAR={1960}, MRNUMBER = {0217083}, ZBLNUMBER = {0118.36206}, }
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@article{EGAII, author={Grothendieck, A.}, MRKEY={0217084}, TITLE={\'{E}léments de géométrie algébrique. {II}. \'{E}tude globale élémentaire de quelques classes de morphismes}, JOURNAL={Inst. Hautes Études Sci. Publ. Math.}, VOLUME={8}, PAGES={222}, YEAR={1961}, MRNUMBER = {0217084}, ZBLNUMBER = {0118.36206}, }
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@article{EGAIII, MRKEY={0217085}, AUTHOR={Grothendieck, A.}, TITLE={\'{E}léments de géométrie algébrique. {III}. \'{E}tude cohomologique des faisceaux cohérents. {I}}, JOURNAL={Inst. Hautes Études Sci. Publ. Math.}, VOLUME={11}, YEAR={1961}, ZBLNUMBER = {0118.36206}, PAGES = {349--511}, MRNUMBER = {0217085}, }
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@article {EGAIV2, MRKEY = {0199181}, AUTHOR = {Grothendieck, A.}, TITLE = {\'{E}léments de géométrie algébrique. {IV}. \'{E}tude locale des schémas et des morphismes de schémas. {II}}, JOURNAL = {Inst. Hautes Études Sci. Publ. Math.}, FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques}, VOLUME = {24}, YEAR = {1965}, ISSN = {0073-8301}, MRCLASS = {14.00}, MRNUMBER = {0199181}, MRREVIEWER = {H. Hironaka}, ZBLNUMBER = {0135.39701}, PAGES = {1--231}, }
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@book {Hart, MRKEY = {0463157}, AUTHOR = {Hartshorne, Robin}, TITLE = {Algebraic Geometry}, SERIES = {Grad. Texts in Math.}, VOLUME={52}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1977}, PAGES = {xvi+496}, ISBN = {0-387-90244-9}, MRCLASS = {14-01}, MRNUMBER = {0463157}, MRREVIEWER = {Robert Speiser}, ZBLNUMBER = {0367.14001}, }
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@book {HTT, MRKEY = {2357361}, AUTHOR = {Hotta, Ryoshi and Takeuchi, Kiyoshi and Tanisaki, Toshiyuki}, TITLE = {{$D$}-Modules, Perverse Sheaves, and Representation Theory}, SERIES = {Progr. Math.}, VOLUME = {236}, NOTE = {translated from the 1995 Japanese edition by Takeuchi}, PUBLISHER = {Birkhäuser}, ADDRESS = {Boston, MA}, YEAR = {2008}, PAGES = {xii+407}, ISBN = {978-0-8176-4363-8}, MRCLASS = {32C38 (14F05 14F10 17B10)}, MRNUMBER = {2357361}, MRREVIEWER = {Corrado Marastoni}, DOI = {10.1007/978-0-8176-4523-6}, ZBLNUMBER = {1136.14009}, }
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@book {Hum, MRKEY = {0323842}, AUTHOR = {Humphreys, James E.}, TITLE = {Introduction to {L}ie Algebras and Representation Theory}, SERIES = {Grad. Texts in Math.}, VOLUME={9}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1972}, PAGES = {xii+169}, MRCLASS = {17BXX}, MRNUMBER = {0323842}, MRREVIEWER = {F. W. Lemire}, ZBLNUMBER = {0254.17004}, DOI = {10.1007/978-1-4612-6398-2}, }
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@book {Hum2, MRKEY = {1343976}, AUTHOR = {Humphreys, James E.}, TITLE = {Conjugacy Classes in Semisimple Algebraic Groups}, SERIES = {Math. Surveys Monogr.}, VOLUME = {43}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {1995}, PAGES = {xviii+196}, ISBN = {0-8218-0333-6}, MRCLASS = {20G15 (22E10)}, MRNUMBER = {1343976}, ZBLNUMBER = {0834.20048}, }
[Noot1] $Go to document$ C. Huyghe, "${\mathcal{D}}^\dagger$-affinité de l’espace projectif," Compositio Math., vol. 108, iss. 3, pp. 277-318, 1997.

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@article {Noot1, MRKEY = {1473850}, AUTHOR = {Huyghe, C.}, TITLE = {{${\mathcal{D}}^\dagger$}-affinité de l'espace projectif}, NOTE = {with an appendix by P. Berthelot}, JOURNAL = {Compositio Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {108}, YEAR = {1997}, NUMBER = {3}, PAGES = {277--318}, ISSN = {0010-437X}, CODEN = {CMPMAF}, MRCLASS = {14F20 (11G99 14F30)}, MRNUMBER = {1473850}, MRREVIEWER = {St{é}phane Guillermou}, DOI = {10.1023/A:1000124232370}, ZBLNUMBER = {0956.14010}, }
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@article {Jaikin, MRKEY = {2169043}, AUTHOR = {Jaikin-Zapirain, A.}, TITLE = {Zeta function of representations of compact {$p$}-adic analytic groups}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {19}, YEAR = {2006}, NUMBER = {1}, PAGES = {91--118}, ISSN = {0894-0347}, MRCLASS = {20E18 (11M41 20C15 22E35)}, MRNUMBER = {2169043}, MRREVIEWER = {Alexander Moret{ó}}, DOI = {10.1090/S0894-0347-05-00501-1}, ZBLNUMBER = {1092.20023}, }
[Ja2] J. C. Jantzen, "Representations of Lie algebras in prime characteristic," in Representation Theories and Algebraic Geometry, Dordrecht: Kluwer Acad. Publ., 1998, vol. 514, pp. 185-235.

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@incollection {Ja2, MRKEY = {1649627}, AUTHOR = {Jantzen, Jens Carsten}, TITLE = {Representations of {L}ie algebras in prime characteristic}, BOOKTITLE = {Representation Theories and Algebraic Geometry}, VENUE={{M}ontreal, {PQ}, 1997}, SERIES = {NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.}, VOLUME = {514}, PAGES = {185--235}, NOTE = {Notes by Iain Gordon}, PUBLISHER = {Kluwer Acad. Publ.}, ADDRESS = {Dordrecht}, YEAR = {1998}, MRCLASS = {17B50}, MRNUMBER = {1649627}, MRREVIEWER = {Daniel K. Nakano}, ZBLNUMBER = {0974.17022}, }
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@book {Jantzen, MRKEY = {2015057}, AUTHOR = {Jantzen, Jens Carsten}, TITLE = {Representations of Algebraic Groups}, SERIES = {Math. Surveys Monogr.}, VOLUME = {107}, EDITION = {Second}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2003}, PAGES = {xiv+576}, ISBN = {0-8218-3527-0}, MRCLASS = {20G05 (17B10)}, MRNUMBER = {2015057}, ZBLNUMBER = {1034.20041}, }
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@book {Lafforgue, MRKEY = {1976905}, AUTHOR = {Lafforgue, L.}, TITLE = {Chirurgie des Grassmanniennes}, SERIES = {CRM Monogr. Ser.}, VOLUME = {19}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2003}, PAGES = {xx+170}, ISBN = {0-8218-3358-8}, MRCLASS = {14M15 (14D20 14N20)}, MRNUMBER = {1976905}, MRREVIEWER = {Harry Tamvakis}, ZBLNUMBER = {1026.14015}, }
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@book {Lang58, MRKEY = {0100591}, AUTHOR = {Lang, Serge}, TITLE = {Introduction to Algebraic Geometry}, PUBLISHER = {Interscience Publishers, New York-London}, YEAR = {1958}, PAGES = {xi+260}, MRCLASS = {14.00}, MRNUMBER = {0100591}, MRREVIEWER = {M. Nagata}, ZBLNUMBER = {0095.15301}, }
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@article {Laz1965, MRKEY = {0209286}, AUTHOR = {Lazard, Michel}, TITLE = {Groupes analytiques {$p$}-adiques}, JOURNAL = {Inst. Hautes Études Sci. Publ. Math.}, FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques}, VOLUME = {26}, YEAR = {1965}, PAGES = {389--603}, ISSN = {0073-8301}, MRCLASS = {14.50}, MRNUMBER = {0209286}, ZBLNUMBER = {0139.02302}, }
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@book {LVO, MRKEY = {1420862}, AUTHOR = {Li, Huishi and van Oystaeyen, Freddy}, TITLE = {Zariskian Filtrations}, SERIES = {$K$ Monographs in Mathematics}, VOLUME = {2}, PUBLISHER = {Kluwer Academic Publishers}, ADDRESS = {Dordrecht}, YEAR = {1996}, PAGES = {x+252}, ISBN = {0-7923-4184-8}, MRCLASS = {16W60 (16-02 16W50)}, MRNUMBER = {1420862}, MRREVIEWER = {Blas Torrecillas Jover}, ZBLNUMBER = {0862.16027}, }
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@article {Li, MRKEY = {1371738}, AUTHOR = {Li, Huishi}, TITLE = {Lifting {O}re sets of {N}oetherian filtered rings and applications}, JOURNAL = {J. Algebra}, FJOURNAL = {Journal of Algebra}, VOLUME = {179}, YEAR = {1996}, NUMBER = {3}, PAGES = {686--703}, ISSN = {0021-8693}, CODEN = {JALGA4}, MRCLASS = {16W60}, MRNUMBER = {1371738}, MRREVIEWER = {Blas Torrecillas Jover}, DOI = {10.1006/jabr.1996.0031}, ZBLNUMBER = {0849.16037}, }
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@book {MacLane, MRKEY = {1712872}, AUTHOR = {Mac Lane, Saunders}, TITLE = {Categories for the Working Mathematician}, SERIES = {Grad. Texts in Math.}, VOLUME = {5}, EDITION = {Second}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1998}, PAGES = {xii+314}, ISBN = {0-387-98403-8}, MRCLASS = {18-02}, MRNUMBER = {1712872}, ZBLNUMBER = {0906.18001}, }
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@misc{MO, author={Liu, Q.}, TITLE={How exotic can dvrs be in the ring of rational functions over a local field?}, NOTE={(mathoverflow.net/users/3485)}, URL={http://mathoverflow.net/questions/49452}, }
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@book {MCR, MRKEY = {1811901}, AUTHOR = {McConnell, J. C. and Robson, J. C.}, TITLE = {Noncommutative {N}oetherian Rings}, SERIES = {Grad. Stud. Math.}, VOLUME = {30}, EDITION = {revised}, NOTE = {with the cooperation of L. W. Small}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2001}, PAGES = {xx+636}, ISBN = {0-8218-2169-5}, MRCLASS = {16P40 (16-02)}, MRNUMBER = {1811901}, ZBLNUMBER = {0980.16019}, }
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@book {Milne, MRKEY = {0559531}, AUTHOR = {Milne, James S.}, TITLE = {\'{E}tale Cohomology}, SERIES = {Princeton Math. Ser.}, VOLUME = {33}, PUBLISHER = {Princeton Univ. Press}, ADDRESS = {Princeton, N.J.}, YEAR = {1980}, PAGES = {xiii+323}, ISBN = {0-691-08238-3}, MRCLASS = {14-02 (14F20 18F99)}, MRNUMBER = {0559531}, MRREVIEWER = {G. Horrocks}, ZBLNUMBER = {0433.14012}, }
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@article {Noot2, MRKEY = {2543473}, AUTHOR = {Noot-Huyghe, Christine}, TITLE = {Un théorème de {B}eilinson-{B}ernstein pour les {$\mathcal{D}$}-modules arithmétiques}, JOURNAL = {Bull. Soc. Math. France}, FJOURNAL = {Bulletin de la Société Mathématique de France}, VOLUME = {137}, YEAR = {2009}, NUMBER = {2}, PAGES = {159--183}, ISSN = {0037-9484}, CODEN = {BSMFAA}, MRCLASS = {14F10 (14G17 14M15)}, MRNUMBER = {2543473}, MRREVIEWER = {Uli Walther}, ZBLNUMBER = {1171.14014}, }
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@book {Pass, MRKEY = {0979094}, AUTHOR = {Passman, Donald S.}, TITLE = {Infinite Crossed Products}, SERIES = {Pure Appl. Math.}, VOLUME = {135}, PUBLISHER = {Academic Press}, ADDRESS = {Boston, MA}, YEAR = {1989}, PAGES = {xii+468}, ISBN = {0-12-546390-1}, MRCLASS = {16-02 (16A03 16A27 20C07)}, MRNUMBER = {0979094}, MRREVIEWER = {Martin Lorenz}, ZBLNUMBER = {0662.16001}, }
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@article {Premet, MRKEY = {1976699}, AUTHOR = {Premet, Alexander}, TITLE = {Nilpotent orbits in good characteristic and the {K}empf-{R}ousseau theory}, NOTE = {special issue celebrating the 80th birthday of Robert Steinberg}, JOURNAL = {J. Algebra}, FJOURNAL = {Journal of Algebra}, VOLUME = {260}, YEAR = {2003}, NUMBER = {1}, PAGES = {338--366}, ISSN = {0021-8693}, CODEN = {JALGA4}, MRCLASS = {17B50 (14L30 20G15)}, MRNUMBER = {1976699}, MRREVIEWER = {George J. McNinch}, DOI = {10.1016/S0021-8693(02)00662-2}, ZBLNUMBER = {1020.20031}, }
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@article {Quillen, MRKEY = {0238892}, AUTHOR = {Quillen, Daniel}, TITLE = {On the endomorphism ring of a simple module over an enveloping algebra}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {21}, YEAR = {1969}, PAGES = {171--172}, ISSN = {0002-9939}, MRCLASS = {16.40 (14.00)}, MRNUMBER = {0238892}, MRREVIEWER = {T. A. Springer}, ZBLNUMBER = {0188.08901}, }
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@misc{Schmidt2, author = {Schmidt, Tobias}, TITLE = {Verma modules over $p$-adic {A}rens--{M}ichael envelopes of reductive {L}ie algebras}, NOTE={preprint}, SORTYEAR={2014}, }
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@article {Schmidt, MRKEY = {2375595}, AUTHOR = {Schmidt, Tobias}, TITLE = {Auslander regularity of {$p$}-adic distribution algebras}, JOURNAL = {Represent. Theory}, FJOURNAL = {Representation Theory. An Electronic Journal of the American Mathematical Society}, VOLUME = {12}, YEAR = {2008}, PAGES = {37--57}, ISSN = {1088-4165}, MRCLASS = {22E50}, MRNUMBER = {2375595}, MRREVIEWER = {Dubravka Ban}, DOI = {10.1090/S1088-4165-08-00323-3}, ZBLNUMBER = {1142.22010}, }
[ST1] $Go to document$ P. Schneider and J. Teitelbaum, "$U({\mathfrak{g}})$-finite locally analytic representations," Represent. Theory, vol. 5, pp. 111-128, 2001.

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@article {ST1, MRKEY = {1835001}, AUTHOR = {Schneider, Peter and Teitelbaum, Jeremy}, TITLE = {{$U({\mathfrak{g}})$}-finite locally analytic representations}, NOTE = {with an appendix by Dipendra Prasad}, JOURNAL = {Represent. Theory}, FJOURNAL = {Representation Theory. An Electronic Journal of the American Mathematical Society}, VOLUME = {5}, YEAR = {2001}, PAGES = {111--128}, ISSN = {1088-4165}, MRCLASS = {22E50 (17B15)}, MRNUMBER = {1835001}, MRREVIEWER = {Marko Tadi{ć}}, DOI = {10.1090/S1088-4165-01-00109-1}, ZBLNUMBER = {1028.17007}, }
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@article {ST2, MRKEY = {1900706}, AUTHOR = {Schneider, Peter and Teitelbaum, Jeremy}, TITLE = {Banach space representations and {I}wasawa theory}, JOURNAL = {Israel J. Math.}, FJOURNAL = {Israel Journal of Mathematics}, VOLUME = {127}, YEAR = {2002}, PAGES = {359--380}, ISSN = {0021-2172}, CODEN = {ISJMAP}, MRCLASS = {22E50}, MRNUMBER = {1900706}, MRREVIEWER = {Dubravka Ban}, DOI = {10.1007/BF02784538}, ZBLNUMBER = {1006.46053}, }
[ST] $Go to document$ P. Schneider and J. Teitelbaum, "Algebras of $p$-adic distributions and admissible representations," Invent. Math., vol. 153, iss. 1, pp. 145-196, 2003.

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@article {ST, MRKEY = {1990669}, AUTHOR = {Schneider, Peter and Teitelbaum, Jeremy}, TITLE = {Algebras of {$p$}-adic distributions and admissible representations}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {153}, YEAR = {2003}, NUMBER = {1}, PAGES = {145--196}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {22E50}, MRNUMBER = {1990669}, MRREVIEWER = {Marko Tadi{ć}}, DOI = {10.1007/s00222-002-0284-1}, ZBLNUMBER = {1028.11070}, }
[ST3] $Go to document$ P. Schneider and J. Teitelbaum, "Duality for admissible locally analytic representations," Represent. Theory, vol. 9, pp. 297-326, 2005.

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On irreducible representations of compact $p$-adic analytic groups

Abstract

Authors