Geometric and homological properties of affine Deligne-Lusztig varieties

Pages 367-404 from Volume 179 (2014), Issue 1 by Xuhua He

Abstract

This paper studies affine Deligne-Lusztig varieties $X_{\tilde w}(b)$ in the affine flag variety of a quasi-split tamely ramified group. We describe the geometric structure of $X_{\tilde w}(b)$ for a minimal length element $\tilde w$ in the conjugacy class of an extended affine Weyl group. We then provide a reduction method that relates the structure of $X_{\tilde w}(b)$ for arbitrary elements $\tilde w$ in the extended affine Weyl group to those associated with minimal length elements. Based on this reduction, we establish a connection between the dimension of affine Deligne-Lusztig varieties and the degree of the class polynomial of affine Hecke algebras. As a consequence, we prove a conjecture of Görtz, Haines, Kottwitz and Reuman.

  • [B1] Go to document R. Bédard, "On the Brauer liftings for modular representations," J. Algebra, vol. 93, iss. 2, pp. 332-353, 1985.
    @article {B1, 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},
      DOI = {10.1016/0021-8693(85)90164-4},
      ZBLNUMBER = {0607.20024},
     }
  • [B] Go to document R. Bédard, "The lowest two-sided cell for an affine Weyl group," Comm. Algebra, vol. 16, iss. 6, pp. 1113-1132, 1988.
    @article {B, MRKEY = {0939034},
      AUTHOR = {B{é}dard, Robert},
      TITLE = {The lowest two-sided cell for an affine {W}eyl group},
      JOURNAL = {Comm. Algebra},
      FJOURNAL = {Communications in Algebra},
      VOLUME = {16},
      YEAR = {1988},
      NUMBER = {6},
      PAGES = {1113--1132},
      ISSN = {0092-7872},
      CODEN = {COALDM},
      MRCLASS = {20H15},
      MRNUMBER = {0939034},
      MRREVIEWER = {Bhama Srinivasan},
      DOI = {10.1080/00927878808823622},
      ZBLNUMBER = {1195.20048},
     }
  • [Bour] N. Bourbaki, Lie Groups and Lie Algebras. Chapters 4–6, New York: Springer-Verlag, 2002.
    @book {Bour, MRKEY = {1890629},
      AUTHOR = {Bourbaki, Nicolas},
      TITLE = {Lie Groups and {L}ie Algebras. {C}hapters 4--6},
      SERIES = {Elements of Math. (Berlin)},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2002},
      PAGES = {xii+300},
      ISBN = {3-540-42650-7},
      MRCLASS = {17-01 (00A05 20E42 20F55 22-01)},
      MRNUMBER = {1890629},
      }
  • [BR08] Go to document C. Bonnafé and R. Rouquier, "Affineness of Deligne-Lusztig varieties for minimal length elements," J. Algebra, vol. 320, iss. 3, pp. 1200-1206, 2008.
    @article {BR08, MRKEY = {2427637},
      AUTHOR = {Bonnaf{é},
      C{é}dric and Rouquier, Rapha{ë}l},
      TITLE = {Affineness of {D}eligne-{L}usztig varieties for minimal length elements},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {320},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {1200--1206},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20G05},
      MRNUMBER = {2427637},
      MRREVIEWER = {Eva Viehmann},
      DOI = {10.1016/j.jalgebra.2007.12.029},
      ZBLNUMBER = {1195.20048},
     }
  • [BT] Go to document F. Bruhat and J. Tits, "Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée," Inst. Hautes Études Sci. Publ. Math., vol. 60, pp. 197-376, 1984.
    @article {BT, MRKEY = {0756316},
      AUTHOR = {Bruhat, F. and Tits, J.},
      TITLE = {Groupes réductifs sur un corps local. {II}. {S}chémas en groupes. {E}xistence d'une donnée radicielle valuée},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {60},
      YEAR = {1984},
      PAGES = {197--376},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {20G25 (14L15)},
      MRNUMBER = {0756316},
      MRREVIEWER = {James E. Humphreys},
      URL = {http://www.numdam.org/item?id=PMIHES_1984__60__5_0},
      ZBLNUMBER = {0597.14041},
     }
  • [DL] Go to document P. Deligne and G. Lusztig, "Representations of reductive groups over finite fields," Ann. of Math., vol. 103, iss. 1, pp. 103-161, 1976.
    @article {DL, MRKEY = {0393266},
      AUTHOR = {Deligne, P. and Lusztig, G.},
      TITLE = {Representations of reductive groups over finite fields},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {103},
      YEAR = {1976},
      NUMBER = {1},
      PAGES = {103--161},
      ISSN = {0003-486X},
      MRCLASS = {20G05 (14M15)},
      MRNUMBER = {0393266},
      MRREVIEWER = {S. I. Gelfand},
      ZBLNUMBER = {0336.20029},
      DOI = {10.2307/1971021},
     }
  • [Ga] Go to document Q. R. Gashi, "On a conjecture of Kottwitz and Rapoport," Ann. Sci. Éc. Norm. Supér., vol. 43, iss. 6, pp. 1017-1038, 2010.
    @article {Ga, MRKEY = {2778454},
      AUTHOR = {Gashi, Q{ë}ndrim R.},
      TITLE = {On a conjecture of {K}ottwitz and {R}apoport},
      JOURNAL = {Ann. Sci. Éc. Norm. Supér.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {43},
      YEAR = {2010},
      NUMBER = {6},
      PAGES = {1017--1038},
      ISSN = {0012-9593},
      MRCLASS = {14L35 (20G15)},
      MRNUMBER = {2778454},
      ZBLNUMBER = {1225.14037},
      URL = {http://smf4.emath.fr/Publications/AnnalesENS/4_43/html/ens_ann-sc_43_1017-1038.php},
     }
  • [GH] U. Görtz and X. He, "Dimensions of affine Deligne-Lusztig varieties in affine flag varieties," Doc. Math., vol. 15, pp. 1009-1028, 2010.
    @article {GH, MRKEY = {2745691},
      AUTHOR = {G{ö}rtz, Ulrich and He, Xuhua},
      TITLE = {Dimensions of affine {D}eligne-{L}usztig varieties in affine flag varieties},
      JOURNAL = {Doc. Math.},
      FJOURNAL = {Documenta Mathematica},
      VOLUME = {15},
      YEAR = {2010},
      PAGES = {1009--1028},
      ISSN = {1431-0635},
      MRCLASS = {20F55 (20G25)},
      MRNUMBER = {2745691},
      MRREVIEWER = {Alan Koch},
      ZBLNUMBER = {1248.20048},
     }
  • [GHKR1] Go to document U. Görtz, T. J. Haines, R. E. Kottwitz, and D. C. Reuman, "Dimensions of some affine Deligne-Lusztig varieties," Ann. Sci. École Norm. Sup., vol. 39, iss. 3, pp. 467-511, 2006.
    @article {GHKR1, MRKEY = {2265676},
      AUTHOR = {G{ö}rtz, Ulrich and Haines, Thomas J. and Kottwitz, Robert E. and Reuman, Daniel C.},
      TITLE = {Dimensions of some affine {D}eligne-{L}usztig varieties},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {39},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {467--511},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {14L15 (14M15 20G25)},
      MRNUMBER = {2265676},
      MRREVIEWER = {Alan Koch},
      DOI = {10.1016/j.ansens.2005.12.004},
      ZBLNUMBER = {1108.14035}
    }
  • [GHKR] 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 {GHKR, 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},
      DOI = {10.1112/S0010437X10004823},
      ZBLNUMBER = {1229.14036},
     }
  • [GHN] U. Görtz, X. He, and S. Nie, ${P}$-alcoves and nonemptiness of affine Deligne-Lusztig varieties.
    @misc{GHN,
      author={Görtz, U. and He, X. and Nie, S.},
      TITLE={${P}$-alcoves and nonemptiness of affine {D}eligne-{L}usztig varieties},
      ARXIV={1211.3784},
     }
  • [GP93] Go to document M. Geck and G. Pfeiffer, "On the irreducible characters of Hecke algebras," Adv. Math., vol. 102, iss. 1, pp. 79-94, 1993.
    @article {GP93, MRKEY = {1250466},
      AUTHOR = {Geck, Meinolf and Pfeiffer, G{ö}tz},
      TITLE = {On the irreducible characters of {H}ecke algebras},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {102},
      YEAR = {1993},
      NUMBER = {1},
      PAGES = {79--94},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {20C15},
      MRNUMBER = {1250466},
      MRREVIEWER = {Hiroshi Naruse},
      DOI = {10.1006/aima.1993.1056},
      }
  • [Ha] T. J. Haines, "Introduction to Shimura varieties with bad reduction of parahoric type," in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Providence, RI: Amer. Math. Soc., 2005, vol. 4, pp. 583-642.
    @incollection {Ha, MRKEY = {2192017},
      AUTHOR = {Haines, Thomas J.},
      TITLE = {Introduction to {S}himura varieties with bad reduction of parahoric type},
      BOOKTITLE = {Harmonic Analysis, the Trace Formula, and {S}himura Varieties},
      SERIES = {Clay Math. Proc.},
      VOLUME = {4},
      PAGES = {583--642},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2005},
      MRCLASS = {11G18 (14G35)},
      MRNUMBER = {2192017},
      MRREVIEWER = {Ulrich G{ö}rtz},
      ZBLNUMBER = {1148.11028},
     }
  • [HR] T. J. Haines and M. Rapoport, On parahoric subgroups.
    @misc{HR,
      author={Haines, Thomas J. and Rapoport, M.},
      TITLE={On parahoric subgroups},
      NOTE={appendix to \cite{PR}},
     }
  • [He072] Go to document X. He, "Minimal length elements in some double cosets of Coxeter groups," Adv. Math., vol. 215, iss. 2, pp. 469-503, 2007.
    @article {He072, MRKEY = {2355597},
      AUTHOR = {He, Xuhua},
      TITLE = {Minimal length elements in some double cosets of {C}oxeter groups},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {215},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {469--503},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {20F55},
      MRNUMBER = {2355597},
      MRREVIEWER = {Andrei Yu. Vesnin},
      DOI = {10.1016/j.aim.2007.04.005},
      ZBLNUMBER = {1149.20035},
     }
  • [H7] Go to document X. He, "Closure of Steinberg fibers and affine Deligne-Lusztig varieties," Int. Math. Res. Not., vol. 2011, iss. 14, pp. 3237-3260, 2011.
    @article {H7, MRKEY = {2817678},
      AUTHOR = {He, Xuhua},
      TITLE = {Closure of {S}teinberg fibers and affine {D}eligne-{L}usztig varieties},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2011},
      NUMBER = {14},
      PAGES = {3237--3260},
      ISSN = {1073-7928},
      MRCLASS = {14L35 (14M99 20G15)},
      MRNUMBER = {2817678},
      MRREVIEWER = {Vladimir L. Popov},
      DOI = {10.1093/imrn/rnq189},
      VOLUME = {2011},
      }
  • [H8] Go to document X. He, "A subalgebra of 0-Hecke algebra," J. Algebra, vol. 322, iss. 11, pp. 4030-4039, 2009.
    @article {H8, MRKEY = {2556136},
      AUTHOR = {He, Xuhua},
      TITLE = {A subalgebra of 0-{H}ecke algebra},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {322},
      YEAR = {2009},
      NUMBER = {11},
      PAGES = {4030--4039},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20C08},
      MRNUMBER = {2556136},
      MRREVIEWER = {Thomas Pietraho},
      DOI = {10.1016/j.jalgebra.2009.04.003},
      ZBLNUMBER = {1186.20005},
     }
  • [He5] X. He, Minimal length elements of extended affine Weyl group, I.
    @misc{He5,
      author={He, Xuhua},
      TITLE = {Minimal length elements of extended affine {W}eyl group, {I}},
      ARXIV={1004.4040},
      SORTYEAR={2014},
     }
  • [HL] Go to document X. He and G. Lusztig, "A generalization of Steinberg’s cross section," J. Amer. Math. Soc., vol. 25, iss. 3, pp. 739-757, 2012.
    @article {HL, MRKEY = {2904572},
      AUTHOR = {He, Xuhua and Lusztig, George},
      TITLE = {A generalization of {S}teinberg's cross section},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {25},
      YEAR = {2012},
      NUMBER = {3},
      PAGES = {739--757},
      ISSN = {0894-0347},
      MRCLASS = {20G15},
      MRNUMBER = {2904572},
      MRREVIEWER = {Toshiyuki Tanisaki},
      DOI = {10.1090/S0894-0347-2012-00728-0},
      ZBLNUMBER = {1252.20047},
     }
  • [HN] X. He and S. Nie, Minimal length elements of extended affine Weyl group, II.
    @misc{HN,
      author={He, Xuhua and Nie, S.},
      TITLE={Minimal length elements of extended affine {W}eyl group, {II}},
      ARXIV={1112.0824},
     }
  • [HW] X. He and T. Wedhorn, On parahoric reductions of Shimura varieties of PEL type.
    @misc{HW,
      author={He, Xuhua and Wedhorn, T.},
      TITLE={On parahoric reductions of {S}himura varieties of {PEL} type},
      NOTE={in preparation},
     }
  • [HY] Go to document X. He and Z. Yang, "Elements with finite Coxeter part in an affine Weyl group," J. Algebra, vol. 372, pp. 204-210, 2012.
    @article {HY, MRKEY = {2990008},
      AUTHOR = {He, Xuhua and Yang, Zhongwei},
      TITLE = {Elements with finite {C}oxeter part in an affine {W}eyl group},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {372},
      YEAR = {2012},
      PAGES = {204--210},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20F55},
      MRNUMBER = {2990008},
      MRREVIEWER = {Sarah B. Hart},
      DOI = {10.1016/j.jalgebra.2012.09.017},
      ZBLNUMBER = {06180107},
     }
  • [HV] 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 {HV, 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},
      DOI = {10.1515/CRELLE.2011.044},
      ZBLNUMBER = {1225.14036},
     }
  • [IM65] N. Iwahori and H. Matsumoto, "On some Bruhat decomposition and the structure of the Hecke rings of ${\germ p}$-adic Chevalley groups," Inst. Hautes Études Sci. Publ. Math., iss. 25, pp. 5-48, 1965.
    @article {IM65, MRKEY = {0185016},
      AUTHOR = {Iwahori, N. and Matsumoto, H.},
      TITLE = {On some {B}ruhat decomposition and the structure of the {H}ecke rings of {${\germ p}$}-adic {C}hevalley groups},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {25},
      YEAR = {1965},
      PAGES = {5--48},
      ISSN = {0073-8301},
      MRCLASS = {20.70 (14.50)},
      MRNUMBER = {0185016},
      MRREVIEWER = {Rimhak Ree},
      }
  • [Ko85] Go to document R. E. Kottwitz, "Isocrystals with additional structure," Compositio Math., vol. 56, iss. 2, pp. 201-220, 1985.
    @article {Ko85, 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},
      URL = {http://www.numdam.org/item?id=CM_1985__56_2_201_0},
      ZBLNUMBER = {0597.20038},
     }
  • [Ko97] Go to document R. E. Kottwitz, "Isocrystals with additional structure. II," Compositio Math., vol. 109, iss. 3, pp. 255-339, 1997.
    @article {Ko97, MRKEY = {1485921},
      AUTHOR = {Kottwitz, Robert E.},
      TITLE = {Isocrystals with additional structure. {II}},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {109},
      YEAR = {1997},
      NUMBER = {3},
      PAGES = {255--339},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {20G25 (11S25 14F30 14L05)},
      MRNUMBER = {1485921},
      MRREVIEWER = {Guy Rousseau},
      DOI = {10.1023/A:1000102604688},
      ZBLNUMBER = {0966.20022},
     }
  • [Ko3] Go to document R. E. Kottwitz, "Dimensions of Newton strata in the adjoint quotient of reductive groups," Pure Appl. Math. Q., vol. 2, iss. 3, Special Issue: In honor of Robert D. MacPherson. Part 1, pp. 817-836, 2006.
    @article {Ko3, MRKEY = {2252119},
      AUTHOR = {Kottwitz, Robert E.},
      TITLE = {Dimensions of {N}ewton strata in the adjoint quotient of reductive groups},
      JOURNAL = {Pure Appl. Math. Q.},
      FJOURNAL = {Pure and Applied Mathematics Quarterly},
      VOLUME = {2},
      YEAR = {2006},
      NUMBER = {3, Special Issue: In honor of Robert D. MacPherson. Part 1},
      PAGES = {817--836},
      ISSN = {1558-8599},
      MRCLASS = {14G35 (11G18 14L05 20G25)},
      MRNUMBER = {2252119},
      MRREVIEWER = {Eva Viehmann},
      DOI = {10.4310/PAMQ.2006.v2.n3.a10},
      ZBLNUMBER = {1109.11033},
     }
  • [KR] Go to document R. E. Kottwitz and M. Rapoport, "On the existence of $F$-crystals," Comment. Math. Helv., vol. 78, iss. 1, pp. 153-184, 2003.
    @article {KR, MRKEY = {1966756},
      AUTHOR = {Kottwitz, Robert E. and Rapoport, M.},
      TITLE = {On the existence of {$F$}-crystals},
      JOURNAL = {Comment. Math. Helv.},
      FJOURNAL = {Commentarii Mathematici Helvetici},
      VOLUME = {78},
      YEAR = {2003},
      NUMBER = {1},
      PAGES = {153--184},
      ISSN = {0010-2571},
      CODEN = {COMHAX},
      MRCLASS = {11S25 (14F30 14L05)},
      MRNUMBER = {1966756},
      MRREVIEWER = {Elmar Grosse-Kl{ö}nne},
      ZBLNUMBER = {1126.14023},
      URL = {http://www.ems-ph.org/journals/show_abstract.php?issn=0010-2571&vol=78&iss=1&rank=8},
      }
  • [K] Go to document D. Krammer, "The conjugacy problem for Coxeter groups," Groups Geom. Dyn., vol. 3, iss. 1, pp. 71-171, 2009.
    @article {K, MRKEY = {2466021},
      AUTHOR = {Krammer, Daan},
      TITLE = {The conjugacy problem for {C}oxeter groups},
      JOURNAL = {Groups Geom. Dyn.},
      FJOURNAL = {Groups, Geometry, and Dynamics},
      VOLUME = {3},
      YEAR = {2009},
      NUMBER = {1},
      PAGES = {71--171},
      ISSN = {1661-7207},
      MRCLASS = {20F55 (20F10)},
      MRNUMBER = {2466021},
      MRREVIEWER = {Robert H. Gilman},
      DOI = {10.4171/GGD/52},
      }
  • [Lu] G. Lusztig, Characters of Reductive Groups over a Finite Field, Princeton, NJ: Princeton Univ. Press, 1984, vol. 107.
    @book {Lu, MRKEY = {0742472},
      AUTHOR = {Lusztig, George},
      TITLE = {Characters of Reductive Groups over a Finite Field},
      SERIES = {Annals of Math. Stud.},
      VOLUME = {107},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1984},
      PAGES = {xxi+384},
      ISBN = {0-691-08350-9; 0-691-08351-7},
      MRCLASS = {20G05 (14L20 20C15)},
      MRNUMBER = {0742472},
      MRREVIEWER = {Bhama Srinivasan},
      ZBLNUMBER = {0556.20033},
     }
  • [Luc] Go to document C. Lucarelli, "A converse to Mazur’s inequality for split classical groups," J. Inst. Math. Jussieu, vol. 3, iss. 2, pp. 165-183, 2004.
    @article {Luc, MRKEY = {2055708},
      AUTHOR = {Lucarelli, Catherine},
      TITLE = {A converse to {M}azur's inequality for split classical groups},
      JOURNAL = {J. Inst. Math. Jussieu},
      FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institute de Mathématiques de Jussieu},
      VOLUME = {3},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {165--183},
      ISSN = {1474-7480},
      MRCLASS = {14L05 (11S25 14F30 20G25)},
      MRNUMBER = {2055708},
      MRREVIEWER = {Alan Koch},
      DOI = {10.1017/S1474748004000064},
      ZBLNUMBER = {1054.14059},
     }
  • [PR] Go to document G. Pappas and M. Rapoport, "Twisted loop groups and their affine flag varieties," Adv. Math., vol. 219, iss. 1, pp. 118-198, 2008.
    @article {PR, MRKEY = {2435422},
      AUTHOR = {Pappas, G. and Rapoport, M.},
      TITLE = {Twisted loop groups and their affine flag varieties},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {219},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {118--198},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {22E67 (14M15 17B67 20G25)},
      MRNUMBER = {2435422},
      MRREVIEWER = {Dmitry A. Timash{ë}v},
      DOI = {10.1016/j.aim.2008.04.006},
      ZBLNUMBER = {1159.22010},
     }
  • [Ra] M. Rapoport, "A guide to the reduction modulo $p$ of Shimura varieties," in Automorphic Forms. I, , 2005, pp. 271-318.
    @incollection {Ra, MRKEY = {2141705},
      AUTHOR = {Rapoport, M.},
      TITLE = {A guide to the reduction modulo {$p$} of {S}himura varieties},
      BOOKTITLE = {Automorphic Forms. I},
      JOURNAL = {Astérisque},
      FJOURNAL = {Astérisque},
      SERIES = {298},
      YEAR = {2005},
      PAGES = {271--318},
      ISSN = {0303-1179},
      MRCLASS = {11G18 (11G40 14G35)},
      MRNUMBER = {2141705},
      MRREVIEWER = {Ulrich G{ö}rtz},
      ZBLNUMBER = {1084.11029},
     }
  • [RR] 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 {RR, 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 = {Abdellah Mokrane},
      URL = {http://www.numdam.org/item?id=CM_1996__103_2_153_0},
      ZBLNUMBER = {0874.14008},
     }
  • [Ri] Go to document T. Richarz, "Schubert varieties in twisted affine flag varieties and local models," J. Algebra, vol. 375, pp. 121-147, 2013.
    @article {Ri, MRKEY = {2998951},
      AUTHOR = {Richarz, Timo},
      TITLE = {Schubert varieties in twisted affine flag varieties and local models},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {375},
      YEAR = {2013},
      PAGES = {121--147},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {14M15},
      MRNUMBER = {2998951},
      DOI = {10.1016/j.jalgebra.2012.11.013},
      }
  • [Sp] Go to document T. A. Springer, "Regular elements of finite reflection groups," Invent. Math., vol. 25, pp. 159-198, 1974.
    @article {Sp, MRKEY = {0354894},
      AUTHOR = {Springer, T. A.},
      TITLE = {Regular elements of finite reflection groups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {25},
      YEAR = {1974},
      PAGES = {159--198},
      ISSN = {0020-9910},
      MRCLASS = {20H15},
      MRNUMBER = {0354894},
      MRREVIEWER = {R. Steinberg},
      ZBLNUMBER = {0287.20043},
      DOI = {10.1007/BF01390173},
      }
  • [Vi] Go to document E. Viehmann, "The dimension of some affine Deligne-Lusztig varieties," Ann. Sci. École Norm. Sup., vol. 39, iss. 3, pp. 513-526, 2006.
    @article {Vi, MRKEY = {2265677},
      AUTHOR = {Viehmann, Eva},
      TITLE = {The dimension of some affine {D}eligne-{L}usztig varieties},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {39},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {513--526},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {14L15 (14M15 20G25)},
      MRNUMBER = {2265677},
      MRREVIEWER = {Alan Koch},
      DOI = {10.1016/j.ansens.2006.04.001},
      ZBLNUMBER = {1108.14036},
     }
  • [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 = {Preliminary Data},
      MRNUMBER = {3072810},
      DOI = {10.1007/s00208-012-0892-z},
      ZBLNUMBER = {06194422},
     }
  • [Zb] Go to document B. Zbarsky, On Some Stratifications of Affine Deligne-Lusztig Varieties for ${ {SL}}(3)$, ProQuest LLC, Ann Arbor, MI, 2008.
    @book {Zb, MRKEY = {2717502},
      AUTHOR = {Zbarsky, Boris},
      TITLE = {On Some Stratifications of Affine {D}eligne-{L}usztig Varieties for ${\rm {SL}}(3)$},
      NOTE = {Ph.D Thesis---The University of Chicago},
      PUBLISHER = {ProQuest LLC, Ann Arbor, MI},
      YEAR = {2008},
      PAGES = {113},
      ISBN = {978-0549-57037-0},
      MRCLASS = {Thesis},
      MRNUMBER = {2717502},
      URL = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3309123},
      }

Keywords
,
DOI
https://doi.org/10.4007/annals.2014.179.1.6
MR
3126571
Zbl
06260590
Milestones
Received: 29 January 2012
Revised: 13 January 2013
Accepted: 26 February 2013
Published online: 1 January 2014

Authors

Xuhua He

The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Current address:

University of Maryland, College Park, MD