Truncations of level $1$ of elements in the loop group of a reductive group

Abstract

The aim of this article is to define and study a new invariant of elements of loop groups that is invariant under $\sigma$-conjugation by a hyperspecial maximal open subgroup and that we call the truncation of level 1. We classify truncations of level 1 and describe their specialization behavior. Furthermore, we prove group-theoretic conditions for the set of $\sigma$-conjugacy classes obtained from elements of a given truncation of level 1 and in particular for the generic $\sigma$-conjugacy class in any given truncation stratum. In the last section we relate our invariant to the Ekedahl-Oort stratification of the Siegel moduli space and to generalizations to other PEL Shimura varieties.

Note: To view the article, click on the URL link for the DOI number.

  • [bedard] Go to document R. Bédard, "On the Brauer liftings for modular representations," J. Algebra, vol. 93, iss. 2, pp. 332-353, 1985.
    @article {bedard, MRKEY = {0786758},
      AUTHOR = {B{é}dard, Robert},
      TITLE = {On the {B}rauer liftings for modular representations},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {93},
      YEAR = {1985},
      NUMBER = {2},
      PAGES = {332--353},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20G05 (20C20)},
      MRNUMBER = {0786758},
      MRREVIEWER = {Bhama Srinivasan},
      ZBLNUMBER = {0607.20024},
      DOI = {10.1016/0021-8693(85)90164-4},
      }
  • [DDPW] B. Deng, J. Du, B. Parshall, and J. Wang, Finite Dimensional Algebras and Quantum Groups, Providence, RI: Amer. Math. Soc., 2008, vol. 150.
    @book {DDPW, MRKEY = {2457938},
      AUTHOR = {Deng, Bangming and Du, Jie and Parshall, Brian and Wang, Jianpan},
      TITLE = {Finite Dimensional Algebras and Quantum Groups},
      SERIES = {Math. Surveys Monogr.},
      VOLUME = {150},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2008},
      PAGES = {xxvi+759},
      ISBN = {978-0-8218-4186-0},
      MRCLASS = {17B37 (05C20 05E10 16G20 17B67 20C08)},
      MRNUMBER = {2457938},
      MRREVIEWER = {Alistair Savage},
      ZBLNUMBER = {1154.17003},
      }
  • [EkedahlvdGeer] Go to document T. Ekedahl and G. van der Geer, "Cycle classes of the E-O stratification on the moduli of abelian varieties," in Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin. Vol. I, Boston, MA: Birkhäuser, 2009, vol. 269, pp. 567-636.
    @incollection {EkedahlvdGeer, MRKEY = {2641181},
      AUTHOR = {Ekedahl, Torsten and van der Geer, Gerard},
      TITLE = {Cycle classes of the {E}-{O} stratification on the moduli of abelian varieties},
      BOOKTITLE = {Algebra, Arithmetic, and Geometry: In Honor of {Y}u. {I}. {M}anin. {V}ol. {I}},
      SERIES = {Progr. Math.},
      VOLUME = {269},
      PAGES = {567--636},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {2009},
      MRCLASS = {14K10 (14K22 14M15 20F55 20G30)},
      MRNUMBER = {2641181},
      MRREVIEWER = {Arvid Siqveland},
      ZBLNUMBER = {1200.14089},
      DOI = {10.1007/978-0-8176-4745-2_13},
      }
  • [Faltings] Go to document G. Faltings, "Algebraic loop groups and moduli spaces of bundles," J. Eur. Math. Soc. $($JEMS$)$, vol. 5, iss. 1, pp. 41-68, 2003.
    @article {Faltings, MRKEY = {1961134},
      AUTHOR = {Faltings, Gerd},
      TITLE = {Algebraic loop groups and moduli spaces of bundles},
      JOURNAL = {J. Eur. Math. Soc. $($JEMS$)$},
      FJOURNAL = {Journal of the European Mathematical Society (JEMS)},
      VOLUME = {5},
      YEAR = {2003},
      NUMBER = {1},
      PAGES = {41--68},
      ISSN = {1435-9855},
      MRCLASS = {14D20 (14M15 17B67 22E67)},
      MRNUMBER = {1961134},
      MRREVIEWER = {Harry Tamvakis},
      ZBLNUMBER = {1020.14002},
      DOI = {10.1007/s10097-002-0045-x},
      }
  • [GHKR2] Go to document U. Görtz, T. J. Haines, R. E. Kottwitz, and D. C. Reuman, "Affine Deligne-Lusztig varieties in affine flag varieties," Compos. Math., vol. 146, iss. 5, pp. 1339-1382, 2010.
    @article {GHKR2, MRKEY = {2684303},
      AUTHOR = {G{ö}rtz, Ulrich and Haines, Thomas J. and Kottwitz, Robert E. and Reuman, Daniel C.},
      TITLE = {Affine {D}eligne-{L}usztig varieties in affine flag varieties},
      JOURNAL = {Compos. Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {146},
      YEAR = {2010},
      NUMBER = {5},
      PAGES = {1339--1382},
      ISSN = {0010-437X},
      MRCLASS = {14L35 (11S25 14F30 20G25)},
      MRNUMBER = {2684303},
      MRREVIEWER = {Alan Koch},
      ZBLNUMBER = {1229.14036},
      DOI = {10.1112/S0010437X10004823},
      }
  • [GY2] Go to document U. Görtz and C. Yu, "The supersingular locus in Siegel modular varieties with Iwahori level structure," Math. Ann., vol. 353, iss. 2, pp. 465-498, 2012.
    @article {GY2, MRKEY = {2915544},
      AUTHOR = {G{ö}rtz, Ulrich and Yu, Chia-Fu},
      TITLE = {The supersingular locus in {S}iegel modular varieties with {I}wahori level structure},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {353},
      YEAR = {2012},
      NUMBER = {2},
      PAGES = {465--498},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {14G35 (11G18)},
      MRNUMBER = {2915544},
      ZBLNUMBER = {1257.14019},
      DOI = {10.1007/s00208-011-0689-5},
      }
  • [H1] Go to document S. Harashita, "Ekedahl-Oort strata and the first Newton slope strata," J. Algebraic Geom., vol. 16, iss. 1, pp. 171-199, 2007.
    @article {H1, MRKEY = {2257323},
      AUTHOR = {Harashita, Shushi},
      TITLE = {Ekedahl-{O}ort strata and the first {N}ewton slope strata},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {16},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {171--199},
      ISSN = {1056-3911},
      MRCLASS = {14G35 (11G18 14K10)},
      MRNUMBER = {2257323},
      MRREVIEWER = {Ulrich G{ö}rtz},
      ZBLNUMBER = {1113.14031},
      DOI = {10.1090/S1056-3911-06-00451-6},
      }
  • [H2] Go to document S. Harashita, "Configuration of the central streams in the moduli of abelian varieties," Asian J. Math., vol. 13, iss. 2, pp. 215-250, 2009.
    @article {H2, MRKEY = {2559109},
      AUTHOR = {Harashita, Shushi},
      TITLE = {Configuration of the central streams in the moduli of abelian varieties},
      JOURNAL = {Asian J. Math.},
      FJOURNAL = {Asian Journal of Mathematics},
      VOLUME = {13},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {215--250},
      ISSN = {1093-6106},
      MRCLASS = {14K10 (11G10)},
      MRNUMBER = {2559109},
      MRREVIEWER = {Nicolae Manolache},
      ZBLNUMBER = {1193.14055},
      DOI = {10.4310/AJM.2009.v13.n2.a4},
      }
  • [H3] Go to document S. Harashita, "Generic Newton polygons of Ekedahl-Oort strata: Oort’s conjecture," Ann. Inst. Fourier $($Grenoble$)$, vol. 60, iss. 5, pp. 1787-1830, 2010.
    @article {H3, MRKEY = {2766230},
      AUTHOR = {Harashita, Shushi},
      TITLE = {Generic {N}ewton polygons of {E}kedahl-{O}ort strata: {O}ort's conjecture},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {60},
      YEAR = {2010},
      NUMBER = {5},
      PAGES = {1787--1830},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {14K05 (13A35 14K10 14L05)},
      MRNUMBER = {2766230},
      MRREVIEWER = {Arvid Siqveland},
      ZBLNUMBER = {1208.14038},
      DOI = {10.5802/aif.2572},
      }
  • [H4] Go to document S. Harashita, "The supremum of Newton polygons of $p$-divisible groups with a given $p$-kernel type," in Geometry and Analysis of Automorphic Forms of Several Variables, World Sci. Publ., Hackensack, NJ, 2012, vol. 7, pp. 41-55.
    @incollection {H4, MRKEY = {2908034},
      AUTHOR = {Harashita, Shushi},
      TITLE = {The supremum of {N}ewton polygons of {$p$}-divisible groups with a given {$p$}-kernel type},
      BOOKTITLE = {Geometry and Analysis of Automorphic Forms of Several Variables},
      SERIES = {Ser. Number Theory Appl.},
      VOLUME = {7},
      PAGES = {41--55},
      PUBLISHER = {World Sci. Publ., Hackensack, NJ},
      YEAR = {2012},
      MRCLASS = {14L05 (14L15)},
      MRNUMBER = {2908034},
      MRREVIEWER = {Eva Viehmann},
      ZBLNUMBER = {06148143},
      DOI = {10.1142/9789814355605_0002},
      }
  • [equidim] Go to document U. Hartl and E. Viehmann, "The Newton stratification on deformations of local $G$-shtukas," J. Reine Angew. Math., vol. 656, pp. 87-129, 2011.
    @article {equidim, MRKEY = {2818857},
      AUTHOR = {Hartl, Urs and Viehmann, Eva},
      TITLE = {The {N}ewton stratification on deformations of local {$G$}-shtukas},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      VOLUME = {656},
      YEAR = {2011},
      PAGES = {87--129},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {14L05 (11G09)},
      MRNUMBER = {2818857},
      MRREVIEWER = {Marc-Hubert Nicole},
      ZBLNUMBER = {1225.14036},
      DOI = {10.1515/CRELLE.2011.044},
      }
  • [foliat] Go to document U. Hartl and E. Viehmann, "Foliations in deformation spaces of local $G$-shtukas," Adv. Math., vol. 229, iss. 1, pp. 54-78, 2012.
    @article {foliat, MRKEY = {2854170},
      AUTHOR = {Hartl, Urs and Viehmann, Eva},
      TITLE = {Foliations in deformation spaces of local {$G$}-shtukas},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {229},
      YEAR = {2012},
      NUMBER = {1},
      PAGES = {54--78},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {20G25 (11G09 14L05)},
      MRNUMBER = {2854170},
      MRREVIEWER = {Dipendra Prasad},
      ZBLNUMBER = {05994235},
      DOI = {10.1016/j.aim.2011.08.011},
      }
  • [He] Go to document X. He, "The $G$-stable pieces of the wonderful compactification," Trans. Amer. Math. Soc., vol. 359, iss. 7, pp. 3005-3024, 2007.
    @article {He, MRKEY = {2299444},
      AUTHOR = {He, Xuhua},
      TITLE = {The {$G$}-stable pieces of the wonderful compactification},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {359},
      YEAR = {2007},
      NUMBER = {7},
      PAGES = {3005--3024},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {14L30 (14M17 20G15)},
      MRNUMBER = {2299444},
      MRREVIEWER = {Simon M. Goodwin},
      ZBLNUMBER = {1124.20033},
      DOI = {10.1090/S0002-9947-07-04158-X},
      }
  • [Kottwitz1] Go to document R. E. Kottwitz, "Isocrystals with additional structure," Compositio Math., vol. 56, iss. 2, pp. 201-220, 1985.
    @article {Kottwitz1, MRKEY = {0809866},
      AUTHOR = {Kottwitz, Robert E.},
      TITLE = {Isocrystals with additional structure},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {56},
      YEAR = {1985},
      NUMBER = {2},
      PAGES = {201--220},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14L25 (14F30 14L15 20G25)},
      MRNUMBER = {0809866},
      MRREVIEWER = {K. F. Lai},
      ZBLNUMBER = {0597.20038},
      URL = {http://www.numdam.org/item?id=CM_1985__56_2_201_0},
      }
  • [Lusztig] G. Lusztig, "Parabolic character sheaves. I," Mosc. Math. J., vol. 4, iss. 1, pp. 153-179, 311, 2004.
    @article {Lusztig, MRKEY = {2074987},
      AUTHOR = {Lusztig, George},
      TITLE = {Parabolic character sheaves. {I}},
      JOURNAL = {Mosc. Math. J.},
      FJOURNAL = {Moscow Mathematical Journal},
      VOLUME = {4},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {153--179, 311},
      ISSN = {1609-3321},
      MRCLASS = {20G99},
      MRNUMBER = {2074987},
      ZBLNUMBER = {1102.20030},
      MRREVIEWER = {Bhama Srinivasan},
      }
  • [MoonenWedhorn] Go to document B. Moonen and T. Wedhorn, "Discrete invariants of varieties in positive characteristic," Int. Math. Res. Not., vol. 2004, p. no. 72, 3855-3903.
    @article {MoonenWedhorn, KEY={MW04},
      SORTYEAR={2012},
      MRKEY = {2104263},
      AUTHOR = {Moonen, Ben and Wedhorn, Torsten},
      TITLE = {Discrete invariants of varieties in positive characteristic},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices},
      VOLUME = {2004},
      PAGES = {No.~72, 3855--3903},
      ISSN = {1073-7928},
      MRCLASS = {14F40 (14G35 14J28 14K10 20G99)},
      MRNUMBER = {2104263},
      MRREVIEWER = {Marco Hien},
      ZBLNUMBER = {1084.14023},
      DOI = {10.1155/S1073792804141263},
      }
  • [EO] Go to document F. Oort, "A stratification of a moduli space of abelian varieties," in Moduli of Abelian Varieties, Basel: Birkhäuser, 2001, vol. 195, pp. 345-416.
    @incollection {EO, MRKEY = {1827027},
      AUTHOR = {Oort, Frans},
      TITLE = {A stratification of a moduli space of abelian varieties},
      BOOKTITLE = {Moduli of Abelian Varieties},
      VENUE={{T}exel {I}sland, 1999},
      SERIES = {Progr. Math.},
      VOLUME = {195},
      PAGES = {345--416},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2001},
      MRCLASS = {14K10 (14L15)},
      MRNUMBER = {1827027},
      MRREVIEWER = {Takashi Ichikawa},
      ZBLNUMBER = {1052.14047},
      DOI = {10.1007/978-3-0348-8303-0_13},
      }
  • [Oort1] Go to document F. Oort, "Foliations in moduli spaces of abelian varieties," J. Amer. Math. Soc., vol. 17, iss. 2, pp. 267-296, 2004.
    @article {Oort1, MRKEY = {2051612},
      AUTHOR = {Oort, Frans},
      TITLE = {Foliations in moduli spaces of abelian varieties},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {17},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {267--296},
      ISSN = {0894-0347},
      MRCLASS = {14K10 (14L05)},
      MRNUMBER = {2051612},
      MRREVIEWER = {G. K. Sankaran},
      ZBLNUMBER = {1041.14018},
      DOI = {10.1090/S0894-0347-04-00449-7},
      }
  • [Oort2] Go to document F. Oort, "Minimal $p$-divisible groups," Ann. of Math., vol. 161, iss. 2, pp. 1021-1036, 2005.
    @article {Oort2, MRKEY = {2153405},
      AUTHOR = {Oort, Frans},
      TITLE = {Minimal {$p$}-divisible groups},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {161},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {1021--1036},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {14L05},
      MRNUMBER = {2153405},
      MRREVIEWER = {R. T. Hoobler},
      ZBLNUMBER = {1081.14065},
      DOI = {10.4007/annals.2005.161.1021},
      }
  • [RapoportRichartz] Go to document M. Rapoport and M. Richartz, "On the classification and specialization of $F$-isocrystals with additional structure," Compositio Math., vol. 103, iss. 2, pp. 153-181, 1996.
    @article {RapoportRichartz, MRKEY = {1411570},
      AUTHOR = {Rapoport, M. and Richartz, M.},
      TITLE = {On the classification and specialization of {$F$}-isocrystals with additional structure},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {103},
      YEAR = {1996},
      NUMBER = {2},
      PAGES = {153--181},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14F30 (22E50)},
      MRNUMBER = {1411570},
      MRREVIEWER = {Abdullah Mokrane},
      ZBLNUMBER = {0874.14008},
      URL = {http://www.numdam.org/item?id=CM_1996__103_2_153_0},
      }
  • [42] Go to document J. Tits, "Reductive groups over local fields," in Automorphic Forms, Representations and $L$-Functions, Providence, R.I.: Amer. Math. Soc., 1979, vol. XXXIII, pp. 29-69.
    @incollection {42, MRKEY = {0546588},
      AUTHOR = {Tits, J.},
      TITLE = {Reductive groups over local fields},
      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},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME={XXXIII},
      PAGES = {29--69},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1979},
      MRCLASS = {20G25 (20G10)},
      MRNUMBER = {0546588},
      ZBLNUMBER = {0415.20035},
      DOI = {10.1090/pspum/033.1},
      }
  • [Vasiu] Go to document A. Vasiu, "Mod $p$ classification of Shimura $F$-crystals," Math. Nachr., vol. 283, iss. 8, pp. 1068-1113, 2010.
    @article {Vasiu, MRKEY = {2680723},
      AUTHOR = {Vasiu, A.},
      TITLE = {Mod {$p$} classification of {S}himura {$F$}-crystals},
      JOURNAL = {Math. Nachr.},
      FJOURNAL = {Mathematische Nachrichten},
      VOLUME = {283},
      YEAR = {2010},
      NUMBER = {8},
      PAGES = {1068--1113},
      ISSN = {0025-584X},
      MRCLASS = {11G10 (11G18 14F30 14L17 14L30)},
      MRNUMBER = {2680723},
      MRREVIEWER = {Eva Viehmann},
      ZBLNUMBER = {1264.11053},
      DOI = {10.1002/mana.200714000},
      }
  • [grothconj] Go to document E. Viehmann, "Newton strata in the loop group of a reductive group," Amer. J. Math., vol. 135, iss. 2, pp. 499-518, 2013.
    @article {grothconj, MRKEY = {3038719},
      AUTHOR = {Viehmann, Eva},
      TITLE = {Newton strata in the loop group of a reductive group},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {135},
      YEAR = {2013},
      NUMBER = {2},
      PAGES = {499--518},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {20G40 (14L05)},
      MRNUMBER = {3038719},
      ZBLNUMBER = {06159562},
      DOI = {10.1353/ajm.2013.0017},
      }
  • [VW] Go to document E. Viehmann and T. Wedhorn, "Ekedahl-Oort and Newton strata for Shimura varieties of PEL type," Math. Ann., vol. 356, iss. 4, pp. 1493-1550, 2013.
    @article {VW, MRKEY = {3072810},
      AUTHOR = {Viehmann, Eva and Wedhorn, Torsten},
      TITLE = {Ekedahl-{O}ort and {N}ewton strata for {S}himura varieties of {PEL} type},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {356},
      YEAR = {2013},
      NUMBER = {4},
      PAGES = {1493--1550},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {14G35},
      MRNUMBER = {3072810},
      ZBLNUMBER = {06194422},
      DOI = {10.1007/s00208-012-0892-z},
      }
  • [Wedhorn] T. Wedhorn, Specialization of ${F}$-zips.
    @misc{Wedhorn,
      author={Wedhorn, T.},
      TITLE={Specialization of ${F}$-zips},
      ARXIV={math/0507175},
      }

Authors

Eva Viehmann

Technische Universität München, Zentrum Mathematik - M11, Boltzmannstr. 3, 85748 Garching, Germany