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

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.

  • [ASZ2] Go to document K. Ajitabh, S. P. Smith, and J. J. Zhang, "Injective resolutions of some regular rings," J. Pure Appl. Algebra, vol. 140, iss. 1, pp. 1-21, 1999.
    @article {ASZ2, MRKEY = {1700566},
      AUTHOR = {Ajitabh, K. and Smith, S. P. and Zhang, J. J.},
      TITLE = {Injective resolutions of some regular rings},
      JOURNAL = {J. Pure Appl. Algebra},
      FJOURNAL = {Journal of Pure and Applied Algebra},
      VOLUME = {140},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {1--21},
      ISSN = {0022-4049},
      CODEN = {JPAAA2},
      MRCLASS = {16E10 (16P40 16W50 18G10)},
      MRNUMBER = {1700566},
      MRREVIEWER = {K. R. Goodearl},
      DOI = {10.1016/S0022-4049(99)00049-3},
      ZBLNUMBER = {0945.16009},
      }
  • [ArdBro2006] K. Ardakov and K. A. Brown, "Ring-theoretic properties of Iwasawa algebras: a survey," Doc. Math., vol. 2006, pp. 7-33.
    @article {ArdBro2006, MRKEY = {2290583},
      AUTHOR = {Ardakov, K. and Brown, K. A.},
      TITLE = {Ring-theoretic properties of {I}wasawa algebras: a survey},
      JOURNAL = {Doc. Math.},
      FJOURNAL = {Documenta Mathematica},
      VOLUME = {2006},
      NOTE = {Extra Vol.},
      PAGES = {7--33},
      ISSN = {1431-0635},
      MRCLASS = {11R23 (16S99)},
      MRNUMBER = {2290583},
      MRREVIEWER = {Francesc Bars},
      ZBLNUMBER = {1125.16011},
      }
  • [ArdThesis] K. Ardakov, Krull dimension of Iwasawa algebras and some related topics, 2004.
    @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},
     }
  • [ArdBro2007] Go to document K. Ardakov and K. A. Brown, "Primeness, semiprimeness and localisation in Iwasawa algebras," Trans. Amer. Math. Soc., vol. 359, iss. 4, pp. 1499-1515, 2007.
    @article {ArdBro2007, MRKEY = {2272136},
      AUTHOR = {Ardakov, Konstantin and Brown, Kenneth A.},
      TITLE = {Primeness, semiprimeness and localisation in {I}wasawa algebras},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {359},
      YEAR = {2007},
      NUMBER = {4},
      PAGES = {1499--1515},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {16L30 (11R23 20C07)},
      MRNUMBER = {2272136},
      MRREVIEWER = {M. E. Keating},
      DOI = {10.1090/S0002-9947-06-04153-5},
      ZBLNUMBER = {1151.16024},
      }
  • [ArdWad2006] Go to document K. Ardakov and S. Wadsley, "Characteristic elements for $p$-torsion Iwasawa modules," J. Algebraic Geom., vol. 15, iss. 2, pp. 339-377, 2006.
    @article {ArdWad2006, MRKEY = {2199061},
      AUTHOR = {Ardakov, Konstantin and Wadsley, Simon},
      TITLE = {Characteristic elements for {$p$}-torsion {I}wasawa modules},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {15},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {339--377},
      ISSN = {1056-3911},
      MRCLASS = {11R23 (11G99)},
      MRNUMBER = {2199061},
      MRREVIEWER = {Francesc Bars},
      DOI = {10.1090/S1056-3911-05-00415-7},
      ZBLNUMBER = {1115.20004},
      }
  • [AMac] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Reading, MA: Addison-Wesley Publishing Co., 1969.
    @book {AMac, MRKEY = {0242802},
      AUTHOR = {Atiyah, M. F. and Macdonald, I. G.},
      TITLE = {Introduction to Commutative Algebra},
      PUBLISHER = {Addison-Wesley Publishing Co.},
      ADDRESS={Reading, MA},
      YEAR = {1969},
      PAGES = {ix+128},
      MRCLASS = {13.00},
      MRNUMBER = {0242802},
      MRREVIEWER = {J. A. Johnson},
      ZBLNUMBER = {0175.03601},
      }
  • [BaRi] Go to document P. Bardsley and R. W. Richardson, "Étale slices for algebraic transformation groups in characteristic $p$," Proc. London Math. Soc., vol. 51, iss. 2, pp. 295-317, 1985.
    @article {BaRi, MRKEY = {0794118},
      AUTHOR = {Bardsley, Peter and Richardson, R. W.},
      TITLE = {\'{E}tale slices for algebraic transformation groups in characteristic {$p$}},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {51},
      YEAR = {1985},
      NUMBER = {2},
      PAGES = {295--317},
      ISSN = {0024-6115},
      CODEN = {PLMTAL},
      MRCLASS = {14L30},
      MRNUMBER = {0794118},
      MRREVIEWER = {Andy R. Magid},
      DOI = {10.1112/plms/s3-51.2.295},
      ZBLNUMBER = {0604.14037},
      }
  • [BB] A. Beuilinson and J. Bernstein, "Localisation de $g$-modules," C. R. Acad. Sci. Paris Sér. I Math., vol. 292, iss. 1, pp. 15-18, 1981.
    @article {BB, MRKEY = {0610137},
      AUTHOR = {Be{\u\i}linson, Alexandre and Bernstein, Joseph},
      TITLE = {Localisation de {$g$}-modules},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
      FJOURNAL = {Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique},
      VOLUME = {292},
      YEAR = {1981},
      NUMBER = {1},
      PAGES = {15--18},
      ISSN = {0151-0509},
      CODEN = {CRSMDY},
      MRCLASS = {14F05 (17B35 20G05)},
      MRNUMBER = {0610137},
      MRREVIEWER = {Floyd L. Williams},
      ZBLNUMBER = {0476.14019},
      }
  • [BB2] A. Beuilinson and J. Bernstein, "A proof of Jantzen conjectures," in I. M. Gel\cprime fand Seminar, Providence, RI: Amer. Math. Soc., 1993, vol. 16, pp. 1-50.
    @incollection {BB2, MRKEY = {1237825},
      AUTHOR = {Be{\u\i}linson, Alexandre and Bernstein, Joseph},
      TITLE = {A proof of {J}antzen conjectures},
      BOOKTITLE = {I. {M}. {G}el\cprime fand {S}eminar},
      SERIES = {Adv. Soviet Math.},
      VOLUME = {16},
      PAGES = {1--50},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1993},
      MRCLASS = {22E47 (14A22 14F05)},
      MRNUMBER = {1237825},
      MRREVIEWER = {D. Mili{\v{c}}i{ć}},
      ZBLNUMBER = {0790.22007},
      }
  • [BeiGinz] Go to document A. Beilinson and V. Ginzburg, "Wall-crossing functors and $\mathcal{D}$-modules," Represent. Theory, vol. 3, pp. 1-31, 1999.
    @article {BeiGinz, MRKEY = {1659527},
      AUTHOR = {Beilinson, Alexandre and Ginzburg, Victor},
      TITLE = {Wall-crossing functors and {$\mathcal{D}$}-modules},
      JOURNAL = {Represent. Theory},
      FJOURNAL = {Representation Theory. An Electronic Journal of the American Mathematical Society},
      VOLUME = {3},
      YEAR = {1999},
      PAGES = {1--31},
      ISSN = {1088-4165},
      MRCLASS = {17B20 (05E99 17B37)},
      MRNUMBER = {1659527},
      MRREVIEWER = {Toshiyuki Tanisaki},
      DOI = {10.1090/S1088-4165-99-00063-1},
      ZBLNUMBER = {0910.05068},
      }
  • [BellFarn] Go to document A. D. Bell and R. Farnsteiner, "On the theory of Frobenius extensions and its application to Lie superalgebras," Trans. Amer. Math. Soc., vol. 335, iss. 1, pp. 407-424, 1993.
    @article {BellFarn, MRKEY = {1097163},
      AUTHOR = {Bell, Allen D. and Farnsteiner, Rolf},
      TITLE = {On the theory of {F}robenius extensions and its application to {L}ie superalgebras},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {335},
      YEAR = {1993},
      NUMBER = {1},
      PAGES = {407--424},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {17B70 (17A70 17B55)},
      MRNUMBER = {1097163},
      MRREVIEWER = {Joris Van der Jeugt},
      DOI = {10.2307/2154275},
      ZBLNUMBER = {0811.16013},
      }
  • [Berth] Go to document P. Berthelot, "${\mathcal{D}}$-modules arithmétiques. I. Opérateurs différentiels de niveau fini," Ann. Sci. École Norm. Sup., vol. 29, iss. 2, pp. 185-272, 1996.
    @article {Berth, MRKEY = {1373933},
      AUTHOR = {Berthelot, Pierre},
      TITLE = {{${\mathcal{D}}$}-modules arithmétiques. {I}. {O}pérateurs différentiels de niveau fini},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {29},
      YEAR = {1996},
      NUMBER = {2},
      PAGES = {185--272},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {14F10 (14F30 14G20)},
      MRNUMBER = {1373933},
      MRREVIEWER = {Johan de Jong},
      URL = {http://www.numdam.org/item?id=ASENS_1996_4_29_2_185_0},
      }
  • [Berth2] P. Berthelot, "Introduction à la théorie arithmétique des $\mathcal{D}$-modules," in Cohomologies $p$-Adiques et Applications Arithmétiques, II, , 2002, vol. 279, pp. 1-80.
    @incollection {Berth2, MRKEY = {1922828},
      AUTHOR = {Berthelot, Pierre},
      TITLE = {Introduction à la théorie arithmétique des {$\mathcal{D}$}-modules},
      BOOKTITLE = {Cohomologies $p$-Adiques et Applications Arithm{é}tiques, II},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {279},
      YEAR = {2002},
      PAGES = {1--80},
      ISSN = {0303-1179},
      MRCLASS = {14F10 (13N10 14F30 14F40 14G22 16S32 32C38)},
      MRNUMBER = {1922828},
      MRREVIEWER = {Elmar Grosse-Kl{ö}nne},
      ZBLNUMBER = {1098.14010},
      }
  • [BMR1] Go to document R. Bezrukavnikov, I. Mirković, and D. Rumynin, "Localization of modules for a semisimple Lie algebra in prime characteristic," Ann. of Math., vol. 167, iss. 3, pp. 945-991, 2008.
    @article {BMR1, MRKEY = {2415389},
      AUTHOR = {Bezrukavnikov, Roman and Mirkovi{ć},
      Ivan and Rumynin, Dmitriy},
      TITLE = {Localization of modules for a semisimple {L}ie algebra in prime characteristic},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {167},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {945--991},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {17B50 (17B10 18E30)},
      MRNUMBER = {2415389},
      MRREVIEWER = {Serge M. Skryabin},
      DOI = {10.4007/annals.2008.167.945},
      ZBLNUMBER = {1220.17009},
      }
  • [Bj89] Go to document J. Björk, "The Auslander condition on noetherian rings," in Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année, New York: Springer-Verlag, 1989, vol. 1404, pp. 137-173.
    @incollection {Bj89, MRKEY = {1035224},
      AUTHOR = {Bj{ö}rk, Jan-Erik},
      TITLE = {The {A}uslander condition on noetherian rings},
      BOOKTITLE = {Séminaire d'{A}lgèbre {P}aul {D}ubreil et {M}arie-{P}aul {M}alliavin, 39ème {A}nnée},
      VENUE={{P}aris, 1987/1988},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1404},
      PAGES = {137--173},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1989},
      MRCLASS = {16A03},
      MRNUMBER = {1035224},
      MRREVIEWER = {Arno van den Essen},
      DOI = {10.1007/BFb0084075},
      ZBLNUMBER = {0696.16006},
      }
  • [Borel] Go to document A. Borel, Linear Algebraic Groups, Second ed., New York: Springer-Verlag, 1991, vol. 126.
    @book {Borel, MRKEY = {1102012},
      AUTHOR = {Borel, Armand},
      TITLE = {Linear Algebraic Groups},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {126},
      EDITION = {Second},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1991},
      PAGES = {xii+288},
      ISBN = {0-387-97370-2},
      MRCLASS = {20-01 (20Gxx)},
      MRNUMBER = {1102012},
      MRREVIEWER = {F. D. Veldkamp},
      DOI = {10.1007/978-1-4612-0941-6},
      ZBLNUMBER = {0726.20030},
      }
  • [BoBrIII] Go to document W. Borho and J. -L. Brylinski, "Differential operators on homogeneous spaces. III. Characteristic varieties of Harish-Chandra modules and of primitive ideals," Invent. Math., vol. 80, iss. 1, pp. 1-68, 1985.
    @article {BoBrIII, MRKEY = {0784528},
      AUTHOR = {Borho, W. and Brylinski, J.-L.},
      TITLE = {Differential operators on homogeneous spaces. {III}. {C}haracteristic varieties of {H}arish-{C}handra modules and of primitive ideals},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {80},
      YEAR = {1985},
      NUMBER = {1},
      PAGES = {1--68},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {22E47 (17B35 32C38 43A85)},
      MRNUMBER = {0784528},
      MRREVIEWER = {D. Mili{\v{c}}i{ć}},
      DOI = {10.1007/BF01388547},
      ZBLNUMBER = {0577.22014},
     }
  • [BoBrII] Go to document W. Borho and J. Brylinski, "Differential operators on homogeneous spaces. II. Relative enveloping algebras," Bull. Soc. Math. France, vol. 117, iss. 2, pp. 167-210, 1989.
    @article {BoBrII, MRKEY = {1015807},
      AUTHOR = {Borho, Walter and Brylinski, Jean-Luc},
      TITLE = {Differential operators on homogeneous spaces. {II}. {R}elative enveloping algebras},
      JOURNAL = {Bull. Soc. Math. France},
      FJOURNAL = {Bulletin de la Société Mathématique de France},
      VOLUME = {117},
      YEAR = {1989},
      NUMBER = {2},
      PAGES = {167--210},
      ISSN = {0037-9484},
      CODEN = {BSMFAA},
      MRCLASS = {17B35 (32C38)},
      MRNUMBER = {1015807},
      MRREVIEWER = {Anthony Joseph},
      URL = {http://www.numdam.org/item?id=BSMF_1989__117_2_167_0},
      ZBLNUMBER = {0702.22019},
     }
  • [BGR] S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry, New York: Springer-Verlag, 1984, vol. 261.
    @book {BGR, MRKEY = {0746961},
      AUTHOR = {Bosch, S. and G{ü}ntzer, U. and Remmert, R.},
      TITLE = {Non-{A}rchimedean Analysis. A Systematic Approach to Rigid Analytic Geometry},
      SERIES = {Grundl. Math. Wissen.},
      VOLUME = {261},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1984},
      PAGES = {xii+436},
      ISBN = {3-540-12546-9},
      MRCLASS = {32K10 (30G05 46P05)},
      MRNUMBER = {0746961},
      MRREVIEWER = {W. Bartenwerfer},
      ZBLNUMBER = {0539.14017},
     }
  • [BoSab] Go to document M. Boyarchenko and M. Sabitova, "The orbit method for profinite groups and a $p$-adic analogue of Brown’s theorem," Israel J. Math., vol. 165, pp. 67-91, 2008.
    @article {BoSab, MRKEY = {2403615},
      AUTHOR = {Boyarchenko, Mitya and Sabitova, Maria},
      TITLE = {The orbit method for profinite groups and a {$p$}-adic analogue of {B}rown's theorem},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {165},
      YEAR = {2008},
      PAGES = {67--91},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {20E18 (14L30 22E60)},
      MRNUMBER = {2403615},
      MRREVIEWER = {Nadia P. Mazza},
      DOI = {10.1007/s11856-008-1004-3},
      ZBLNUMBER = {1165.20022},
      }
  • [Caro] Go to document D. Caro, "Holonomie sans structure de Frobenius et critères d’holonomie," Ann. Inst. Fourier (Grenoble), vol. 61, iss. 4, pp. 1437-1454, 2011.
    @article{Caro, MRKEY = {2951498},
      AUTHOR = {Caro, Daniel},
      TITLE = {Holonomie sans structure de {F}robenius et critères d'holonomie},
      JOURNAL = {Ann. Inst. Fourier (Grenoble)},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {61},
      YEAR = {2011},
      NUMBER = {4},
      PAGES = {1437--1454},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {14F10 (14F30)},
      MRNUMBER = {2951498},
      ZBLNUMBER = {06018605},
      MRREVIEWER = {Corrado Marastoni},
      DOI = {10.5802/aif.2645},
      }
  • [CFKSV] Go to document J. Coates, T. Fukaya, K. Kato, R. Sujatha, and O. Venjakob, "The $ GL_2$ main conjecture for elliptic curves without complex multiplication," Publ. Math. Inst. Hautes Études Sci., vol. 101, pp. 163-208, 2005.
    @article {CFKSV, MRKEY = {2217048},
      AUTHOR = {Coates, John and Fukaya, Takako and Kato, Kazuya and Sujatha, Ramdorai and Venjakob, Otmar},
      TITLE = {The {$\rm GL\sb 2$} main conjecture for elliptic curves without complex multiplication},
      JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
      FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      VOLUME = {101},
      YEAR = {2005},
      PAGES = {163--208},
      ISSN = {0073-8301},
      MRCLASS = {11R23 (11G40)},
      MRNUMBER = {2217048},
      MRREVIEWER = {Th{\cfac{o}}ng Nguy{\cftil{e}}n-Quang-{\Dbar}{\cftil{o}}},
      DOI = {10.1007/s10240-004-0029-3},
      ZBLNUMBER = {1108.11081},
      }
  • [CoaSuj1999] Go to document J. Coates and R. Sujatha, "Euler-Poincaré characteristics of abelian varieties," C. R. Acad. Sci. Paris Sér. I Math., vol. 329, iss. 4, pp. 309-313, 1999.
    @article {CoaSuj1999, MRKEY = {1713337},
      AUTHOR = {Coates, John and Sujatha, Ramdorai},
      TITLE = {Euler-{P}oincaré characteristics of abelian varieties},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
      FJOURNAL = {Comptes Rendus de l'Académie des Sciences. Série I. Mathématique},
      VOLUME = {329},
      YEAR = {1999},
      NUMBER = {4},
      PAGES = {309--313},
      ISSN = {0764-4442},
      CODEN = {CASMEI},
      MRCLASS = {11G10 (11R34 14K15)},
      MRNUMBER = {1713337},
      MRREVIEWER = {Antoine Chambert-Loir},
      DOI = {10.1016/S0764-4442(00)88572-9},
      ZBLNUMBER = {0967.14029},
     }
  • [CM] D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, New York: Van Nostrand Reinhold Co., 1993.
    @book {CM, MRKEY = {1251060},
      AUTHOR = {Collingwood, David H. and McGovern, William M.},
      TITLE = {Nilpotent Orbits in Semisimple {L}ie Algebras},
      SERIES = {Van Nostrand Reinhold Math. Series},
      PUBLISHER = {Van Nostrand Reinhold Co.},
      ADDRESS = {New York},
      YEAR = {1993},
      PAGES = {xiv+186},
      ISBN = {0-534-18834-6},
      MRCLASS = {17-02 (17B20 17B25 22E60)},
      MRNUMBER = {1251060},
      MRREVIEWER = {Stephen Slebarski},
      ZBLNUMBER = {0972.17008},
      }
  • [Del] Go to document P. Deligne, "Catégories tannakiennes," in The Grothendieck Festschrift, Vol. II, Boston, MA: Birkhäuser, 1990, vol. 87, pp. 111-195.
    @incollection {Del, MRKEY = {1106898},
      AUTHOR = {Deligne, P.},
      TITLE = {Catégories tannakiennes},
      BOOKTITLE = {The {G}rothendieck {F}estschrift, {V}ol. {II}},
      SERIES = {Progr. Math.},
      VOLUME = {87},
      PAGES = {111--195},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1990},
      MRCLASS = {14A99 (12H05 18A99)},
      MRNUMBER = {1106898},
      MRREVIEWER = {James Milne},
      ZBLNUMBER = {0727.14010},
      DOI = {10.1007/978-0-8176-4575-5_3},
      }
  • [Dema] Go to document M. Demazure, "Invariants symétriques entiers des groupes de Weyl et torsion," Invent. Math., vol. 21, pp. 287-301, 1973.
    @article {Dema, MRKEY = {0342522},
      AUTHOR = {Demazure, Michel},
      TITLE = {Invariants symétriques entiers des groupes de {W}eyl et torsion},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {21},
      YEAR = {1973},
      PAGES = {287--301},
      ISSN = {0020-9910},
      MRCLASS = {14M15 (20G05 20H15 32M10)},
      MRNUMBER = {0342522},
      MRREVIEWER = {S. I. Gelfand},
      ZBLNUMBER = {0269.22010},
      DOI = {10.1007/BF01418790},
     }
  • [Dix] J. Dixmier, Enveloping Algebras, Providence, RI: Amer. Math. Soc., 1996, vol. 11.
    @book {Dix, MRKEY = {1393197},
      AUTHOR = {Dixmier, Jacques},
      TITLE = {Enveloping Algebras},
      SERIES = {Grad. Stud. Math.},
      VOLUME = {11},
      NOTE = {revised reprint of the 1977 translation},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1996},
      PAGES = {xx+379},
      ISBN = {0-8218-0560-6},
      MRCLASS = {17B35},
      MRNUMBER = {1393197},
      ZBLNUMBER = {0867.17001},
      }
  • [DDMS] Go to document J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$ groups, Second ed., Cambridge: Cambridge Univ. Press, 1999, vol. 61.
    @book {DDMS, MRKEY = {1720368},
      AUTHOR = {Dixon, J. D. and du Sautoy, M. P. F. and Mann, A. and Segal, D.},
      TITLE = {Analytic pro-{$p$} groups},
      SERIES = {Cambridge Stud. Adv. Math.},
      VOLUME = {61},
      EDITION = {Second},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1999},
      PAGES = {xviii+368},
      ISBN = {0-521-65011-9},
      MRCLASS = {20E18 (20G30)},
      MRNUMBER = {1720368},
      MRREVIEWER = {Alexander Lubotzky},
      DOI = {10.1017/CBO9780511470882},
      ZBLNUMBER = {0934.20001},
      }
  • [DouRoh] Go to document M. J. Douglass and G. Röhrle, "The Steinberg variety and representations of reductive groups," J. Algebra, vol. 321, iss. 11, pp. 3158-3196, 2009.
    @article {DouRoh, MRKEY = {2510045},
      AUTHOR = {Douglass, J. Matthew and R{ö}hrle, Gerhard},
      TITLE = {The {S}teinberg variety and representations of reductive groups},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {321},
      YEAR = {2009},
      NUMBER = {11},
      PAGES = {3158--3196},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20G05 (14M15 17B08 20C08)},
      MRNUMBER = {2510045},
      MRREVIEWER = {Ben Webster},
      DOI = {10.1016/j.jalgebra.2008.10.027},
      ZBLNUMBER = {1183.20044},
      }
  • [Feit] W. Feit, The Representation Theory of Finite Groups, Amsterdam: North-Holland Publishing Co., 1982, vol. 25.
    @book {Feit, MRKEY = {0661045},
      AUTHOR = {Feit, Walter},
      TITLE = {The Representation Theory of Finite Groups},
      SERIES = {North-Holland Math. Library},
      VOLUME = {25},
      PUBLISHER = {North-Holland Publishing Co.},
      ADDRESS = {Amsterdam},
      YEAR = {1982},
      PAGES = {xiv+502},
      ISBN = {0-444-86155-6},
      MRCLASS = {20-02 (20C20)},
      MRNUMBER = {0661045},
      MRREVIEWER = {J. L. Alperin},
      ZBLNUMBER = {0493.20007},
      }
  • [FvdPut] J. Fresnel and M. van der Put, Rigid Analytic Geometry and its Applications, Boston, MA: Birkhäuser, 2004, vol. 218.
    @book {FvdPut, MRKEY = {2014891},
      AUTHOR = {Fresnel, Jean and van der Put, Marius},
      TITLE = {Rigid Analytic Geometry and its Applications},
      SERIES = {Progr. Math.},
      VOLUME = {218},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {2004},
      PAGES = {xii+296},
      ISBN = {0-8176-4206-4},
      MRCLASS = {14G22 (30G06 30H05 32P05)},
      MRNUMBER = {2014891},
      MRREVIEWER = {Lorenzo Ramero},
      ZBLNUMBER = {1096.14014},
      }
  • [Frommer] Go to document H. Frommer, The locally analytic principal series of split reductive groups, 2003.
    @misc{Frommer,
      author={Frommer, H.},
      TITLE={The locally analytic principal series of split reductive groups},
      NOTE={Münster: SFB-preprint},
      PAGES={265},
      YEAR={2003},
      URL = {http://www. math.uni-muenster.de/sfb/about/publ/heft265.ps},
     }
  • [Gabber] Go to document O. Gabber, "The integrability of the characteristic variety," Amer. J. Math., vol. 103, iss. 3, pp. 445-468, 1981.
    @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},
      }
  • [Gabriel] P. Gabriel, "Des catégories abéliennes," Bull. Soc. Math. France, vol. 90, pp. 323-448, 1962.
    @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},
      }
  • [GK] I. M. Gelcprimefand and A. A. Kirillov, "Structure of the Lie sfield connected with a semisimple decomposable Lie algebra," Funkcional. Anal. i Priložen., vol. 3, iss. 1, pp. 7-26, 1969.
    @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},
      }
  • [Ginz] Go to document V. Ginsburg, "Characteristic varieties and vanishing cycles," Invent. Math., vol. 84, iss. 2, pp. 327-402, 1986.
    @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.
    @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},
      }
  • [EGAI] A. Grothendieck, "Éléments de géométrie algébrique. I. Le langage des schémas," Inst. Hautes Études Sci. Publ. Math., vol. 4, pp. 1-228, 1960.
    @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},
      }
  • [EGAII] A. Grothendieck, "Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes," Inst. Hautes Études Sci. Publ. Math., vol. 8, p. 222, 1961.
    @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},
      }
  • [EGAIII] A. Grothendieck, "Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I," Inst. Hautes Études Sci. Publ. Math., vol. 11, pp. 349-511, 1961.
    @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},
     }
  • [EGAIV2] A. Grothendieck, "Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II," Inst. Hautes Études Sci. Publ. Math., vol. 24, pp. 1-231, 1965.
    @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},
      }
  • [Hart] R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977, vol. 52.
    @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},
      }
  • [HTT] Go to document R. Hotta, K. Takeuchi, and T. Tanisaki, $D$-Modules, Perverse Sheaves, and Representation Theory, Boston, MA: Birkhäuser, 2008, vol. 236.
    @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},
      }
  • [Hum] Go to document J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, New York: Springer-Verlag, 1972, vol. 9.
    @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},
     }
  • [Hum2] J. E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, Providence, RI: Amer. Math. Soc., 1995, vol. 43.
    @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.
    @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},
      }
  • [Jaikin] Go to document A. Jaikin-Zapirain, "Zeta function of representations of compact $p$-adic analytic groups," J. Amer. Math. Soc., vol. 19, iss. 1, pp. 91-118, 2006.
    @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.
    @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},
      }
  • [Jantzen] J. C. Jantzen, Representations of Algebraic Groups, Second ed., Providence, RI: Amer. Math. Soc., 2003, vol. 107.
    @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},
      }
  • [KW] Go to document V. Kac and B. Weisfeiler, "Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic $p$," Indag. Math., vol. 38, iss. 2, pp. 136-151, 1976.
    @article {KW, MRKEY = {0417308},
      AUTHOR = {Kac, V. and Weisfeiler, B.},
      TITLE = {Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic {$p$}},
      JOURNAL = {Indag. Math.},
      VOLUME = {38},
      YEAR = {1976},
      NUMBER = {2},
      PAGES = {136--151},
      MRCLASS = {20G15},
      MRNUMBER = {0417308},
      MRREVIEWER = {F. D. Veldkamp},
      ZBLNUMBER = {0324.17001},
      DOI = {10.1016/1385-7258(76)90059-7},
     }
  • [Lafforgue] L. Lafforgue, Chirurgie des Grassmanniennes, Providence, RI: Amer. Math. Soc., 2003, vol. 19.
    @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},
      }
  • [Lang58] S. Lang, Introduction to Algebraic Geometry, Interscience Publishers, New York-London, 1958.
    @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},
      }
  • [Laz1965] M. Lazard, "Groupes analytiques $p$-adiques," Inst. Hautes Études Sci. Publ. Math., vol. 26, pp. 389-603, 1965.
    @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},
      }
  • [Lev91] Go to document T. Levasseur, "Some properties of noncommutative regular graded rings," Glasgow Math. J., vol. 34, iss. 3, pp. 277-300, 1992.
    @article {Lev91, MRKEY = {1181768},
      AUTHOR = {Levasseur, Thierry},
      TITLE = {Some properties of noncommutative regular graded rings},
      JOURNAL = {Glasgow Math. J.},
      FJOURNAL = {Glasgow Mathematical Journal},
      VOLUME = {34},
      YEAR = {1992},
      NUMBER = {3},
      PAGES = {277--300},
      ISSN = {0017-0895},
      CODEN = {GLMJAS},
      MRCLASS = {16P90 (16W50)},
      MRNUMBER = {1181768},
      MRREVIEWER = {Michel Van den Bergh},
      DOI = {10.1017/S0017089500008843},
      }
  • [LVO] H. Li and F. van Oystaeyen, Zariskian Filtrations, Dordrecht: Kluwer Academic Publishers, 1996, vol. 2.
    @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},
      }
  • [Li] Go to document H. Li, "Lifting Ore sets of Noetherian filtered rings and applications," J. Algebra, vol. 179, iss. 3, pp. 686-703, 1996.
    @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},
      }
  • [MacLane] S. Mac Lane, Categories for the Working Mathematician, Second ed., New York: Springer-Verlag, 1998, vol. 5.
    @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},
      }
  • [MO] Go to document Q. Liu, How exotic can dvrs be in the ring of rational functions over a local field?.
    @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},
      }
  • [MCR] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, revised ed., Providence, RI: Amer. Math. Soc., 2001, vol. 30.
    @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},
      }
  • [Milne] J. S. Milne, Étale Cohomology, Princeton, N.J.: Princeton Univ. Press, 1980, vol. 33.
    @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},
      }
  • [Noot2] C. Noot-Huyghe, "Un théorème de Beilinson-Bernstein pour les $\mathcal{D}$-modules arithmétiques," Bull. Soc. Math. France, vol. 137, iss. 2, pp. 159-183, 2009.
    @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},
      }
  • [Pangalos] Go to document A. Pangalos, $p$-adic Weyl algebras, 2007.
    @misc{Pangalos,
      author={Pangalos, A.},
      TITLE={$p$-adic {W}eyl algebras},
      NOTE={{Ph.D.} thesis, University of Münster},
      YEAR={2007},
      URL = {http://d-nb.info/987679023/34},
     }
  • [Pass] D. S. Passman, Infinite Crossed Products, Boston, MA: Academic Press, 1989, vol. 135.
    @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},
      }
  • [Premet] Go to document A. Premet, "Nilpotent orbits in good characteristic and the Kempf-Rousseau theory," J. Algebra, vol. 260, iss. 1, pp. 338-366, 2003.
    @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},
      }
  • [Quillen] D. Quillen, "On the endomorphism ring of a simple module over an enveloping algebra," Proc. Amer. Math. Soc., vol. 21, pp. 171-172, 1969.
    @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},
      }
  • [Schmidt2] T. Schmidt, Verma modules over $p$-adic Arens–Michael envelopes of reductive Lie algebras.
    @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},
     }
  • [Schmidt] Go to document T. Schmidt, "Auslander regularity of $p$-adic distribution algebras," Represent. Theory, vol. 12, pp. 37-57, 2008.
    @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.
    @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},
      }
  • [ST2] Go to document P. Schneider and J. Teitelbaum, "Banach space representations and Iwasawa theory," Israel J. Math., vol. 127, pp. 359-380, 2002.
    @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.
    @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.
    @article {ST3, MRKEY = {2133762},
      AUTHOR = {Schneider, Peter and Teitelbaum, Jeremy},
      TITLE = {Duality for admissible locally analytic representations},
      JOURNAL = {Represent. Theory},
      FJOURNAL = {Representation Theory. An Electronic Journal of the American Mathematical Society},
      VOLUME = {9},
      YEAR = {2005},
      PAGES = {297--326},
      ISSN = {1088-4165},
      MRCLASS = {22E50},
      MRNUMBER = {2133762},
      MRREVIEWER = {Anne-Marie H. Aubert},
      DOI = {10.1090/S1088-4165-05-00277-3},
      ZBLNUMBER = {1146.22301},
      }
  • [SchVen] Go to document P. Schneider and O. Venjakob, ${SK}_1$ and Lie algebras, 2011.
    @misc{SchVen,
      author = {Schneider, Peter and Venjakob, O.},
      TITLE={${SK}_1$ and {L}ie algebras},
      URL={http://www.math.uni-muenster.de/u/pschnei/publ/pre/SK_1-final.pdf},
      YEAR={2011},
     }
  • [Serre] Go to document . J-P. Serre, Linear representations of finite groups, New York: Springer-Verlag, 1977.
    @book {Serre, MRKEY = {0450380},
      AUTHOR = {Serre, {\relax J-P}},
      TITLE = {Linear representations of finite groups},
      NOTE = {translated from the second French edition by Leonard L. Scott, \emph{Grad. Texts in Math.},
      {\bf 42}},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {x+170},
      ISBN = {0-387-90190-6},
      MRCLASS = {20CXX},
      MRNUMBER = {0450380},
      MRREVIEWER = {W. Feit},
      ZBLNUMBER = {0355.20006},
      DOI = {10.1007/978-1-4684-9458-7},
     }
  • [Smith] Go to document S. P. Smith, "Krull dimension of factor rings of the enveloping algebra of a semi simple Lie algebra," Math. Proc. Cambridge Philos. Soc., vol. 93, iss. 3, pp. 459-466, 1983.
    @article {Smith, MRKEY = {0698349},
      AUTHOR = {Smith, S. P.},
      TITLE = {Krull dimension of factor rings of the enveloping algebra of a semi simple {L}ie algebra},
      JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
      FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
      VOLUME = {93},
      YEAR = {1983},
      NUMBER = {3},
      PAGES = {459--466},
      ISSN = {0305-0041},
      CODEN = {MPCPCO},
      MRCLASS = {17B35 (17B20)},
      MRNUMBER = {0698349},
      DOI = {10.1017/S030500410006076X},
      ZBLNUMBER = {0535.17008},
     }
  • [Soi1] Go to document Y. Soibelman, "On non-commutative analytic spaces over non-Archimedean fields," in Homological Mirror Symmetry, New York: Springer-Verlag, 2009, vol. 757, pp. 221-247.
    @incollection {Soi1, MRKEY = {2596639},
      AUTHOR = {Soibelman, Yan},
      TITLE = {On non-commutative analytic spaces over non-{A}rchimedean fields},
      BOOKTITLE = {Homological Mirror Symmetry},
      SERIES = {Lecture Notes in Phys.},
      VOLUME = {757},
      PAGES = {221--247},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2009},
      MRCLASS = {14G22 (14A22 14J33)},
      MRNUMBER = {2596639},
      MRREVIEWER = {Mark Gross},
      ZBLNUMBER = {1166.81025},
      DOI = {10.1007/978-3-540-68030-7_7},
      }
  • [Soi2] Go to document Y. Soibelman, "Quantum $p$-adic spaces and quantum $p$-adic groups," in Geometry and Dynamics of Groups and Spaces, Basel: Birkhäuser, 2008, vol. 265, pp. 697-719.
    @incollection {Soi2, MRKEY = {2402420},
      AUTHOR = {Soibelman, Yan},
      TITLE = {Quantum {$p$}-adic spaces and quantum {$p$}-adic groups},
      BOOKTITLE = {Geometry and Dynamics of Groups and Spaces},
      SERIES = {Progr. Math.},
      VOLUME = {265},
      PAGES = {697--719},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2008},
      MRCLASS = {12J25 (14J32 17B37)},
      MRNUMBER = {2402420},
      MRREVIEWER = {Jorge Plazas},
      DOI = {10.1007/978-3-7643-8608-5_18},
      ZBLNUMBER = {1139.12008},
      }
  • [SZ] Go to document J. T. Stafford and J. J. Zhang, "Homological properties of (graded) Noetherian ${ PI}$ rings," J. Algebra, vol. 168, iss. 3, pp. 988-1026, 1994.
    @article {SZ, MRKEY = {1293638},
      AUTHOR = {Stafford, J. T. and Zhang, J. J.},
      TITLE = {Homological properties of (graded) {N}oetherian {${\rm PI}$} rings},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {168},
      YEAR = {1994},
      NUMBER = {3},
      PAGES = {988--1026},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {16R10 (16E10 16P40 16W50)},
      MRNUMBER = {1293638},
      MRREVIEWER = {C. N{\u{a}}st{\u{a}}sescu},
      DOI = {10.1006/jabr.1994.1267},
      ZBLNUMBER = {0812.16046},
      }
  • [Stein] Go to document R. Steinberg, "On the desingularization of the unipotent variety," Invent. Math., vol. 36, pp. 209-224, 1976.
    @article {Stein, MRKEY = {0430094},
      AUTHOR = {Steinberg, Robert},
      TITLE = {On the desingularization of the unipotent variety},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {36},
      YEAR = {1976},
      PAGES = {209--224},
      ISSN = {0020-9910},
      MRCLASS = {20G15 (14E15 14M15)},
      MRNUMBER = {0430094},
      MRREVIEWER = {S. I. Gelfand},
      ZBLNUMBER = {0352.20035},
      DOI = {10.1007/BF01390010},
     }
  • [V] Go to document O. Venjakob, "On the structure theory of the Iwasawa algebra of a $p$-adic Lie group," J. Eur. Math. Soc. $($JEMS$)$, vol. 4, iss. 3, pp. 271-311, 2002.
    @article {V, MRKEY = {1924402},
      AUTHOR = {Venjakob, Otmar},
      TITLE = {On the structure theory of the {I}wasawa algebra of a {$p$}-adic {L}ie group},
      JOURNAL = {J. Eur. Math. Soc. $($JEMS$)$},
      FJOURNAL = {Journal of the European Mathematical Society (JEMS)},
      VOLUME = {4},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {271--311},
      ISSN = {1435-9855},
      MRCLASS = {16S34 (22E35)},
      MRNUMBER = {1924402},
      DOI = {10.1007/s100970100038},
      ZBLNUMBER = {1049.16016},
      }
  • [WeiK] Go to document C. A. Weibel, The K-book: An introduction to algebraic K-theory, 2011.
    @misc{WeiK,
      author={Weibel, C. A.},
      TITLE={The {K}-book: An introduction to algebraic {K}-theory},
      URL={http://www.math.rutgers.edu/~weibel/Kbook.html},
      YEAR={2011},
     }
  • [YZ] Go to document A. Yekutieli and J. J. Zhang, "Rings with Auslander dualizing complexes," J. Algebra, vol. 213, iss. 1, pp. 1-51, 1999.
    @article {YZ, MRKEY = {1674648},
      AUTHOR = {Yekutieli, Amnon and Zhang, James J.},
      TITLE = {Rings with {A}uslander dualizing complexes},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {213},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {1--51},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {16E30},
      MRNUMBER = {1674648},
      MRREVIEWER = {Peter Jørgensen},
      DOI = {10.1006/jabr.1998.7657},
      ZBLNUMBER = {0948.16006},
      }

Authors

Konstantin Ardakov

School of Mathematical Sciences, Queen Mary, University of London, MIle End Road, London E1 4NS, United Kingdom

Simon Wadsley

Homerton College, Hills Road, Cambridge CB2 8PH, United Kingdom