Affine Grassmannians and the geometric Satake in mixed characteristic

Abstract

We endow the set of lattices in $\mathbb{Q}_p^n$ with a reasonable algebro-geometric structure. As a result, we prove the representability of affine Grassmannians and establish the geometric Satake equivalence in mixed characteristic. We also give an application of our theory to the study of Rapoport-Zink spaces.

  • [Ap] Go to document J. Alper, "Adequate moduli spaces and geometrically reductive group schemes," Algebr. Geom., vol. 1, iss. 4, pp. 489-531, 2014.
    @ARTICLE{Ap, mrkey = {3272912},
      number = {4},
      issn = {2214-2584},
      author = {Alper, Jarod},
      mrclass = {14D20 (14A20 14L15 14L24 14L30)},
      doi = {10.14231/AG-2014-022},
      journal = {Algebr. Geom.},
      zblnumber = {1322.14026},
      volume = {1},
      mrnumber = {3272912},
      fjournal = {Algebraic Geometry},
      mrreviewer = {Fabio Tonini},
      title = {Adequate moduli spaces and geometrically reductive group schemes},
      year = {2014},
      pages = {489--531},
      }
  • [BL] Go to document A. Beauville and Y. Laszlo, "Conformal blocks and generalized theta functions," Comm. Math. Phys., vol. 164, iss. 2, pp. 385-419, 1994.
    @ARTICLE{BL, mrkey = {1289330},
      number = {2},
      issn = {0010-3616},
      author = {Beauville, Arnaud and Laszlo, Yves},
      mrclass = {14D20 (14H60 17B67)},
      journal = {Comm. Math. Phys.},
      zblnumber = {0815.14015},
      volume = {164},
      mrnumber = {1289330},
      fjournal = {Communications in Mathematical Physics},
      mrreviewer = {Emma Previato},
      coden = {CMPHAY},
      title = {Conformal blocks and generalized theta functions},
      year = {1994},
      pages = {385--419},
      doi = {10.1007/BF02101707},
      }
  • [BD] Go to document A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves.
    @MISC{BD,
      author = {Beilinson, A. and Drinfeld, V.},
      note = {preprint},
      url = {http://www.math.uchicago.edu/~mitya/langlands/QuantizationHitchin.pdf},
      title = {Quantization of {H}itchin's integrable system and {H}ecke eigensheaves},
      }
  • [BS] B. Bhatt and P. Scholze, Projectivity of Witt vector Grassmannians.
    @MISC{BS,
      author = {Bhatt, B. and Scholze, P.},
      note = {preprint},
      arxiv = {1507.06490},
      title = {Projectivity of {W}itt vector {G}rassmannians},
      }
  • [CKV] Go to document M. Chen, M. Kisin, and E. Viehmann, "Connected components of affine Deligne-Lusztig varieties in mixed characteristic," Compos. Math., vol. 151, iss. 9, pp. 1697-1762, 2015.
    @ARTICLE{CKV, mrkey = {3406443},
      number = {9},
      issn = {0010-437X},
      author = {Chen, Miaofen and Kisin, Mark and Viehmann, Eva},
      mrclass = {14G35 (20G25)},
      doi = {10.1112/S0010437X15007253},
      journal = {Compos. Math.},
      zblnumber = {1334.14017},
      volume = {151},
      mrnumber = {3406443},
      fjournal = {Compositio Mathematica},
      mrreviewer = {Zongbin Chen},
      title = {Connected components of affine {D}eligne-{L}usztig varieties in mixed characteristic},
      year = {2015},
      pages = {1697--1762},
      }
  • [CLO] Go to document B. Conrad, M. Lieblich, and M. Olsson, "Nagata compactification for algebraic spaces," J. Inst. Math. Jussieu, vol. 11, iss. 4, pp. 747-814, 2012.
    @ARTICLE{CLO, mrkey = {2979821},
      number = {4},
      issn = {1474-7480},
      author = {Conrad, Brian and Lieblich, Max and Olsson, Martin},
      mrclass = {14A20 (14A15)},
      doi = {10.1017/S1474748011000223},
      journal = {J. Inst. Math. Jussieu},
      zblnumber = {1255.14003},
      volume = {11},
      mrnumber = {2979821},
      fjournal = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu},
      mrreviewer = {Sylvain Brochard},
      title = {Nagata compactification for algebraic spaces},
      year = {2012},
      pages = {747--814},
      }
  • [C] Go to document B. Conrad, The Keel-Mori theorem via stacks.
    @misc{C,
      author={Conrad, B.},
      TITLE={The {K}eel-{M}ori theorem via stacks},
      URL={http://math.stanford.edu/%7Econrad/papers/coarsespace.pdf},
      }
  • [DG] Go to document V. Drinfeld and D. Gaitsgory, "On a theorem of Braden," Transform. Groups, vol. 19, iss. 2, pp. 313-358, 2014.
    @ARTICLE{DG, mrkey = {3200429},
      number = {2},
      issn = {1083-4362},
      author = {Drinfeld, V. and Gaitsgory, D.},
      mrclass = {14F10},
      doi = {10.1007/s00031-014-9267-8},
      journal = {Transform. Groups},
      zblnumber = {1314.14039},
      volume = {19},
      mrnumber = {3200429},
      fjournal = {Transformation Groups},
      mrreviewer = {Christian Schnell},
      title = {On a theorem of {B}raden},
      year = {2014},
      pages = {313--358},
      }
  • [EGA4] Go to document A. Grothendieck, "Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III," Inst. Hautes Études Sci. Publ. Math., vol. 28, iss. 28, p. 255, 1966.
    @ARTICLE{EGA4, mrkey = {0217086},
      number = {28},
      issn = {0073-8301},
      author = {Grothendieck, A.},
      mrclass = {14.55},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      volume = {28},
      zblnumber = {0144.19904},
      mrnumber = {0217086},
      fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      mrreviewer = {J. P. Murre},
      title = {\'{E}léments de géométrie algébrique. {IV}. \'{E}tude locale des schémas et des morphismes de schémas. {III}},
      year = {1966},
      pages = {255},
      doi = {10.1007/BF02684343},
     }
  • [EGA4-4] A. Grothendieck, "Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV," Inst. Hautes Études Sci. Publ. Math., vol. 32, iss. 32, p. 361, 1967.
    @ARTICLE{EGA4-4, mrkey = {0238860},
      number = {32},
      issn = {0073-8301},
      author = {Grothendieck, A.},
      mrclass = {14.55},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      volume = {32},
      zblnumber = {0153.22301},
      mrnumber = {0238860},
      fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      mrreviewer = {J. P. Murre},
      title = {\'{E}léments de géométrie algébrique. {IV}. \'{E}tude locale des schémas et des morphismes de schémas {IV}},
      year = {1967},
      pages = {361},
      }
  • [Ga] Go to document D. Gaitsgory, "Construction of central elements in the affine Hecke algebra via nearby cycles," Invent. Math., vol. 144, iss. 2, pp. 253-280, 2001.
    @ARTICLE{Ga, mrkey = {1826370},
      number = {2},
      issn = {0020-9910},
      author = {Gaitsgory, D.},
      mrclass = {14L15 (14C25 14F05 14M15)},
      doi = {10.1007/s002220100122},
      journal = {Invent. Math.},
      zblnumber = {1072.14055},
      volume = {144},
      mrnumber = {1826370},
      fjournal = {Inventiones Mathematicae},
      mrreviewer = {Burt Totaro},
      coden = {INVMBH},
      title = {Construction of central elements in the affine {H}ecke algebra via nearby cycles},
      year = {2001},
      pages = {253--280},
      }
  • [Gi1] Go to document V. A. Ginzburg, "Sheaves on a loop group, and Langlands duality," Funktsional. Anal. i Prilozhen., vol. 24, iss. 4, pp. 76-77, 1990.
    @ARTICLE{Gi1, mrkey = {1092806},
      number = {4},
      issn = {0374-1990},
      author = {Ginzburg, V. A.},
      mrclass = {22E67 (18F20)},
      doi = {10.1007/BF01077338},
      journal = {Funktsional. Anal. i Prilozhen.},
      volume = {24},
      mrnumber = {1092806},
      fjournal = {Akademiya Nauk SSSR. Funktsional\cprime nyĭAnaliz i ego Prilozheniya},
      mrreviewer = {Helmut Boseck},
      title = {Sheaves on a loop group, and {L}anglands duality},
      year = {1990},
      pages = {76--77},
      zblnumber = {0736.22009},
      }
  • [GHKR] 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{GHKR, mrkey = {2265676},
      number = {3},
      issn = {0012-9593},
      author = {G{ö}rtz, Ulrich and Haines, Thomas J. and Kottwitz, Robert E. and Reuman, Daniel C.},
      mrclass = {14L15 (14M15 20G25)},
      doi = {10.1016/j.ansens.2005.12.004},
      journal = {Ann. Sci. École Norm. Sup.},
      zblnumber = {1108.14035},
      volume = {39},
      mrnumber = {2265676},
      fjournal = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      mrreviewer = {Alan Koch},
      coden = {ASENAH},
      title = {Dimensions of some affine {D}eligne-{L}usztig varieties},
      year = {2006},
      pages = {467--511},
      }
  • [Grb] Go to document M. J. Greenberg, "Schemata over local rings," Ann. of Math., vol. 73, pp. 624-648, 1961.
    @ARTICLE{Grb, mrkey = {0126449},
      issn = {0003-486X},
      author = {Greenberg, Marvin J.},
      mrclass = {14.40},
      journal = {Ann. of Math.},
      zblnumber = {0115.39004},
      volume = {73},
      mrnumber = {0126449},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {P. Cartier},
      title = {Schemata over local rings},
      year = {1961},
      pages = {624--648},
      doi = {10.2307/1970321},
      }
  • [Ha] Go to document W. J. Haboush, "Infinite dimensional algebraic geometry: algebraic structures on $p$-adic groups and their homogeneous spaces," Tohoku Math. J., vol. 57, iss. 1, pp. 65-117, 2005.
    @ARTICLE{Ha, mrkey = {2113991},
      number = {1},
      issn = {0040-8735},
      author = {Haboush, William J.},
      mrclass = {14A15 (14L15 20G25)},
      journal = {Tohoku Math. J.},
      zblnumber = {1119.14004},
      volume = {57},
      mrnumber = {2113991},
      fjournal = {The Tohoku Mathematical Journal. Second Series},
      mrreviewer = {Andy R. Magid},
      coden = {TOMJAM},
      title = {Infinite dimensional algebraic geometry: algebraic structures on {$p$}-adic groups and their homogeneous spaces},
      year = {2005},
      pages = {65--117},
      doi = {10.2748/tmj/1113234835},
      }
  • [HR] T. Haines and M. Rapoport, On parahoric subgroups.
    @MISC{HR,
      author = {Haines, T. and Rapoport, M.},
      note = {appendix to [PR]},
      title = {On parahoric subgroups},
      }
  • [Ham1] Go to document P. Hamacher, "The geometry of Newton strata in the reduction modulo $p$ of Shimura varieties of PEL type," Duke Math. J., vol. 164, iss. 15, pp. 2809-2895, 2015.
    @ARTICLE{Ham1, mrkey = {3430453},
      number = {15},
      issn = {0012-7094},
      author = {Hamacher, Paul},
      mrclass = {14G35 (14L05 20G25)},
      doi = {10.1215/00127094-3328137},
      journal = {Duke Math. J.},
      zblnumber = {1335.14008},
      volume = {164},
      mrnumber = {3430453},
      fjournal = {Duke Mathematical Journal},
      mrreviewer = {Eva Viehmann},
      title = {The geometry of {N}ewton strata in the reduction modulo {$p$} of {S}himura varieties of {PEL} type},
      year = {2015},
      pages = {2809--2895},
      }
  • [Ham2] Go to document P. Hamacher, "The dimension of affine Deligne-Lusztig varieties in the affine Grassmannian," Int. Math. Res. Not., vol. 2015, iss. 23, pp. 12804-12839, 2015.
    @ARTICLE{Ham2, mrkey = {3431637},
      number = {23},
      issn = {1073-7928},
      author = {Hamacher, Paul},
      mrclass = {14L35},
      doi = {10.1093/imrn/rnv081},
      journal = {Int. Math. Res. Not.},
      zblnumber = {06527279},
      volume = {2015},
      mrnumber = {3431637},
      fjournal = {International Mathematics Research Notices. IMRN},
      mrreviewer = {Anthony Henderson},
      title = {The dimension of affine {D}eligne-{L}usztig varieties in the affine {G}rassmannian},
      year = {2015},
      pages = {12804--12839},
      }
  • [HP] B. Howard and G. Pappas, Rapoport-Zink spaces for spinor groups, 2015.
    @MISC{HP,
      author = {Howard, B. and Pappas, G.},
      arxiv = {1509.03914},
      title = {Rapoport-{Z}ink spaces for spinor groups},
      year = {2015},
      }
  • [It] Go to document S. Itoh, "On weak normality and symmetric algebras," J. Algebra, vol. 85, iss. 1, pp. 40-50, 1983.
    @ARTICLE{It, mrkey = {0723066},
      number = {1},
      issn = {0021-8693},
      author = {Itoh, Shiroh},
      mrclass = {13B20 (13B25)},
      doi = {10.1016/0021-8693(83)90117-5},
      journal = {J. Algebra},
      zblnumber = {0534.13002},
      volume = {85},
      mrnumber = {0723066},
      fjournal = {Journal of Algebra},
      mrreviewer = {Douglas L. Costa},
      coden = {JALGA4},
      title = {On weak normality and symmetric algebras},
      year = {1983},
      pages = {40--50},
      }
  • [Ka] N. M. Katz, "Slope filtration of $F$-crystals," in Journées de Géométrie Algébrique de Rennes, Vol. I, Paris: Soc. Math. France, 1979, vol. 63, pp. 113-163.
    @INCOLLECTION{Ka, mrkey = {0563463},
      author = {Katz, Nicholas M.},
      mrclass = {14F30 (14G20 14L05)},
      series = {Astérisque},
      address = {Paris},
      publisher = {Soc. Math. France},
      zblnumber = {0426.14007},
      volume = {63},
      mrnumber = {0563463},
      booktitle = {Journées de {G}éométrie {A}lgébrique de {R}ennes, {V}ol. {I}},
      mrreviewer = {Noriko Yui},
      venue = {{R}ennes, 1978},
      title = {Slope filtration of {$F$}-crystals},
      pages = {113--163},
      year = {1979},
      }
  • [KaLu] Go to document D. Kazhdan and G. Lusztig, "Representations of Coxeter groups and Hecke algebras," Invent. Math., vol. 53, iss. 2, pp. 165-184, 1979.
    @ARTICLE{KaLu, mrkey = {0560412},
      number = {2},
      issn = {0020-9910},
      author = {Kazhdan, David and Lusztig, George},
      mrclass = {20H15 (17B35 20G05 22E47)},
      doi = {10.1007/BF01390031},
      journal = {Invent. Math.},
      zblnumber = {0499.20035},
      volume = {53},
      mrnumber = {0560412},
      fjournal = {Inventiones Mathematicae},
      mrreviewer = {Vinay V. Deodhar},
      coden = {INVMBH},
      title = {Representations of {C}oxeter groups and {H}ecke algebras},
      year = {1979},
      pages = {165--184},
      }
  • [KaLu2] D. Kazhdan and G. Lusztig, "Schubert varieties and Poincaré duality," in Geometry of the Laplace Operator, Providence, RI: Amer. Math. Soc., 1980, vol. XXXVI, pp. 185-203.
    @INCOLLECTION{KaLu2, mrkey = {0573434},
      author = {Kazhdan, David and Lusztig, George},
      mrclass = {14M15 (55N35 57R19)},
      series = {Proc. Sympos. Pure Math.},
      address = {Providence, RI},
      publisher = {Amer. Math. Soc.},
      zblnumber = {0461.14015},
      volume = {XXXVI},
      mrnumber = {0573434},
      booktitle = {Geometry of the {L}aplace Operator},
      venue = {{P}roc. {S}ympos. {P}ure {M}ath., {U}niv. {H}awaii, {H}onolulu, {H}awaii, 1979},
      title = {Schubert varieties and {P}oincaré duality},
      pages = {185--203},
      year = {1980},
      }
  • [KM] Go to document S. Keel and S. Mori, "Quotients by groupoids," Ann. of Math., vol. 145, iss. 1, pp. 193-213, 1997.
    @ARTICLE{KM, mrkey = {1432041},
      number = {1},
      issn = {0003-486X},
      author = {Keel, Se{á}n and Mori, Shigefumi},
      mrclass = {14D25 (14L30)},
      doi = {10.2307/2951828},
      journal = {Ann. of Math.},
      zblnumber = {0881.14018},
      volume = {145},
      mrnumber = {1432041},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {Andrzej Bia{\l}ynicki-Birula},
      coden = {ANMAAH},
      title = {Quotients by groupoids},
      year = {1997},
      pages = {193--213},
      }
  • [Ki] W. Kim, Rapoport-Zink spaces of Hodge type, 2013.
    @MISC{Ki,
      author = {Kim, W.},
      arxiv = {1308.5537},
      title = {Rapoport-{Z}ink spaces of {H}odge type},
      year = {2013},
      }
  • [Kn] Go to document D. Knutson, Algebraic Spaces, New York: Springer-Verlag, 1971, vol. 203.
    @BOOK{Kn, mrkey = {0302647},
      author = {Knutson, Donald},
      mrclass = {14A20 (13J15 14F20)},
      series = {Lecture Notes in Math.},
      address = {New York},
      publisher = {Springer-Verlag},
      zblnumber = {0221.14001},
      volume = {203},
      mrnumber = {0302647},
      mrreviewer = {H. Kurke},
      title = {Algebraic Spaces},
      year = {1971},
      pages = {vi+261},
      doi = {10.1007/BFb0059750},
      }
  • [Ko] Go to document R. E. Kottwitz, "Isocrystals with additional structure. II," Compositio Math., vol. 109, iss. 3, pp. 255-339, 1997.
    @ARTICLE{Ko, mrkey = {1485921},
      number = {3},
      issn = {0010-437X},
      author = {Kottwitz, Robert E.},
      mrclass = {20G25 (11S25 14F30 14L05)},
      doi = {10.1023/A:1000102604688},
      journal = {Compositio Math.},
      zblnumber = {0966.20022},
      volume = {109},
      mrnumber = {1485921},
      fjournal = {Compositio Mathematica},
      mrreviewer = {Guy Rousseau},
      coden = {CMPMAF},
      title = {Isocrystals with additional structure. {II}},
      year = {1997},
      pages = {255--339},
      }
  • [Kr] Go to document M. Kreidl, "On $p$-adic lattices and Grassmannians," Math. Z., vol. 276, iss. 3-4, pp. 859-888, 2014.
    @ARTICLE{Kr, mrkey = {3175163},
      number = {3-4},
      issn = {0025-5874},
      author = {Kreidl, Martin},
      mrclass = {14L15 (13F35 14C05 14G17 14L35)},
      doi = {10.1007/s00209-013-1225-y},
      journal = {Math. Z.},
      zblnumber = {1304.13010},
      volume = {276},
      mrnumber = {3175163},
      fjournal = {Mathematische Zeitschrift},
      mrreviewer = {Hendrik Verhoek},
      title = {On {$p$}-adic lattices and {G}rassmannians},
      year = {2014},
      pages = {859--888},
      }
  • [Laf] V. Lafforgue, Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale, 2012.
    @MISC{Laf,
      author = {Lafforgue, V.},
      arxiv = {1209.5352},
      title = {Chtoucas pour les groupes réductifs et paramétrisation de {L}anglands globale},
      year = {2012},
      }
  • [La] Go to document E. Lau, "Smoothness of the truncated display functor," J. Amer. Math. Soc., vol. 26, iss. 1, pp. 129-165, 2013.
    @ARTICLE{La, mrkey = {2983008},
      number = {1},
      issn = {0894-0347},
      author = {Lau, Eike},
      mrclass = {14F30 (14L05)},
      doi = {10.1090/S0894-0347-2012-00744-9},
      journal = {J. Amer. Math. Soc.},
      zblnumber = {1273.14040},
      volume = {26},
      mrnumber = {2983008},
      fjournal = {Journal of the American Mathematical Society},
      mrreviewer = {Eva Viehmann},
      title = {Smoothness of the truncated display functor},
      year = {2013},
      pages = {129--165},
      }
  • [LMB] G. Laumon and L. Moret-Bailly, Champs Algébriques, New York: Springer-Verlag, 2000, vol. 39.
    @BOOK{LMB, mrkey = {1771927},
      author = {Laumon, G{é}rard and Moret-Bailly, Laurent},
      mrclass = {14A20 (14D20)},
      series = {Ergeb. Math. Grenzgeb.},
      address = {New York},
      isbn = {3-540-65761-4},
      publisher = {Springer-Verlag},
      volume = {39},
      mrnumber = {1771927},
      mrreviewer = {Dan Edidin},
      title = {Champs Algébriques},
      year = {2000},
      pages = {xii+208},
      zblnumber = {0945.14005},
      }
  • [Lu] G. Lusztig, "Singularities, character formulas, and a $q$-analog of weight multiplicities," in Analysis and Topology on Singular Spaces, II, III, Paris: Soc. Math. France, 1983, vol. 101, pp. 208-229.
    @INCOLLECTION{Lu, mrkey = {0737932},
      author = {Lusztig, George},
      mrclass = {17B10 (05A30 20G05 22E47)},
      series = {Astérisque},
      address = {Paris},
      publisher = {Soc. Math. France},
      zblnumber = {0561.22013},
      volume = {101},
      mrnumber = {0737932},
      booktitle = {Analysis and Topology on Singular Spaces, {II},
      {III}},
      mrreviewer = {James E. Humphreys},
      venue = {{L}uminy, 1981},
      title = {Singularities, character formulas, and a {$q$}-analog of weight multiplicities},
      pages = {208--229},
      year = {1983},
      }
  • [Lu2] G. Lusztig, "A bar operator for involutions in a Coxeter group," Bull. Inst. Math. Acad. Sin., vol. 7, iss. 3, pp. 355-404, 2012.
    @ARTICLE{Lu2, mrkey = {3051318},
      number = {3},
      issn = {2304-7909},
      author = {Lusztig, George},
      mrclass = {20F55},
      journal = {Bull. Inst. Math. Acad. Sin.},
      zblnumber = {1283.20045},
      volume = {7},
      mrnumber = {3051318},
      fjournal = {Bulletin of the Institute of Mathematics. Academia Sinica. New Series},
      title = {A bar operator for involutions in a {C}oxeter group},
      year = {2012},
      pages = {355--404},
      }
  • [LV] Go to document G. Lusztig and D. A. Vogan Jr., "Hecke algebras and involutions in Weyl groups," Bull. Inst. Math. Acad. Sin., vol. 7, iss. 3, pp. 323-354, 2012.
    @ARTICLE{LV, mrkey = {3051317},
      number = {3},
      issn = {2304-7909},
      author = {Lusztig, George and Vogan, Jr., David A.},
      mrclass = {20C08 (20G40)},
      journal = {Bull. Inst. Math. Acad. Sin.},
      zblnumber = {1288.20006},
      volume = {7},
      mrnumber = {3051317},
      fjournal = {Bulletin of the Institute of Mathematics. Academia Sinica. New Series},
      mrreviewer = {Anthony Henderson},
      title = {Hecke algebras and involutions in {W}eyl groups},
      year = {2012},
      pages = {323--354},
      URL ={http://web.math.sinica.edu.tw/bulletin/archives_articlecontent16.jsp?bid=MjAxMjMwMQ==},
     }
  • [LY] G. Lusztig and Z. Yun, "A $(-q)$-analogue of weight multiplicities," J. Ramanujan Math. Soc., vol. 28A, pp. 311-340, 2013.
    @ARTICLE{LY, mrkey = {3115198},
      issn = {0970-1249},
      author = {Lusztig, George and Yun, Zhiwei},
      mrclass = {20G05 (14D24)},
      journal = {J. Ramanujan Math. Soc.},
      zblnumber = {1295.20046},
      volume = {28A},
      mrnumber = {3115198},
      fjournal = {Journal of the Ramanujan Mathematical Society},
      mrreviewer = {Anthony Henderson},
      title = {A {$(-q)$}-analogue of weight multiplicities},
      year = {2013},
      pages = {311--340},
      }
  • [Ma] Go to document M. Manaresi, "Some properties of weakly normal varieties," Nagoya Math. J., vol. 77, pp. 61-74, 1980.
    @ARTICLE{Ma, mrkey = {0556308},
      issn = {0027-7630},
      author = {Manaresi, Mirella},
      mrclass = {13H10 (14M05)},
      journal = {Nagoya Math. J.},
      zblnumber = {0403.14001},
      volume = {77},
      mrnumber = {0556308},
      fjournal = {Nagoya Mathematical Journal},
      mrreviewer = {Nadia Chiarli},
      coden = {NGMJA2},
      title = {Some properties of weakly normal varieties},
      year = {1980},
      pages = {61--74},
      doi = {10.1017/S0027763000018663},
      }
  • [MV] Go to document I. Mirković and K. Vilonen, "Geometric Langlands duality and representations of algebraic groups over commutative rings," Ann. of Math., vol. 166, iss. 1, pp. 95-143, 2007.
    @ARTICLE{MV, mrkey = {2342692},
      number = {1},
      issn = {0003-486X},
      author = {Mirkovi{ć},
      I. and Vilonen, K.},
      mrclass = {22E55 (11R39 20G05)},
      doi = {10.4007/annals.2007.166.95},
      journal = {Ann. of Math.},
      zblnumber = {1138.22013},
      volume = {166},
      mrnumber = {2342692},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {Peter Fiebig},
      coden = {ANMAAH},
      title = {Geometric {L}anglands duality and representations of algebraic groups over commutative rings},
      year = {2007},
      pages = {95--143},
      }
  • [NP] B. C. Ngô and P. Polo, "Résolutions de Demazure affines et formule de Casselman-Shalika géométrique," J. Algebraic Geom., vol. 10, iss. 3, pp. 515-547, 2001.
    @ARTICLE{NP, mrkey = {1832331},
      number = {3},
      issn = {1056-3911},
      author = {Ng{ô},
      B. C. and Polo, P.},
      mrclass = {14F43 (14F17)},
      journal = {J. Algebraic Geom.},
      zblnumber = {1041.14002},
      volume = {10},
      mrnumber = {1832331},
      fjournal = {Journal of Algebraic Geometry},
      mrreviewer = {Nguy{\cftil{e}}n Qu{ô}c Th{á}ng},
      title = {Résolutions de {D}emazure affines et formule de {C}asselman-{S}halika géométrique},
      year = {2001},
      pages = {515--547},
      }
  • [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},
      number = {1},
      issn = {0001-8708},
      author = {Pappas, G. and Rapoport, M.},
      mrclass = {22E67 (14M15 17B67 20G25)},
      doi = {10.1016/j.aim.2008.04.006},
      journal = {Adv. Math.},
      zblnumber = {1159.22010},
      volume = {219},
      mrnumber = {2435422},
      fjournal = {Advances in Mathematics},
      mrreviewer = {Dmitry A. Timash{ë}v},
      coden = {ADMTA4},
      title = {Twisted loop groups and their affine flag varieties},
      year = {2008},
      pages = {118--198},
      }
  • [R] M. Rapoport, "A guide to the reduction modulo $p$ of Shimura varieties," in Automorphic Forms. I, Paris: Soc. Math. France, 2005, vol. 298, pp. 271-318.
    @INCOLLECTION{R, mrkey = {2141705},
      issn = {0303-1179},
      author = {Rapoport, Michael},
      mrclass = {11G18 (11G40 14G35)},
      series = {Astérisque},
      address = {Paris},
      publisher = {Soc. Math. France},
      zblnumber = {1084.11029},
      volume = {298},
      mrnumber = {2141705},
      booktitle = {Automorphic Forms. I},
      mrreviewer = {Ulrich G{ö}rtz},
      title = {A guide to the reduction modulo {$p$} of {S}himura varieties},
      year = {2005},
      pages = {271--318},
      }
  • [RZ] Go to document M. Rapoport and T. Zink, Period Spaces for $p$-Divisible Groups, Princeton, NJ: Princeton Univ. Press, 1996, vol. 141.
    @BOOK{RZ, mrkey = {1393439},
      author = {Rapoport, Michael and Zink, Th.},
      mrclass = {14G20 (11G18 14F30 14L05 14M15 20G05 20G25)},
      series = {Annals of Math. Stud.},
      isbn = {0-691-02782-X; 0-691-02781-1},
      address = {Princeton, NJ},
      publisher = {Princeton Univ. Press},
      doi = {10.1515/9781400882601},
      zblnumber = {0873.14039},
      volume = {141},
      mrnumber = {1393439},
      mrreviewer = {Robert E. Kottwitz},
      title = {Period Spaces for {$p$}-Divisible Groups},
      year = {1996},
      pages = {xxii+324},
      }
  • [SW] Go to document P. Scholze and J. Weinstein, $p$-adic geometry.
    @MISC{SW,
      author = {Scholze, P. and Weinstein, J.},
      note = {{L}ecture notes from course at {UC B}erkeley in {F}all 2014},
      url = {http://math.bu.edu/people/jsweinst/Math274/ScholzeLectures.pdf},
      title = {$p$-adic geometry},
      }
  • [Se] Go to document . J-P. Serre, "Groupes proalgébriques," Inst. Hautes Études Sci. Publ. Math., vol. 7, iss. 7, p. 5, 1960.
    @ARTICLE{Se, mrkey = {0118722},
      number = {7},
      issn = {0073-8301},
      author = {Serre, {\relax J-P}},
      mrclass = {14.00},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      volume = {7},
      mrnumber = {0118722},
      fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      mrreviewer = {P. Cartier},
      title = {Groupes proalgébriques},
      year = {1960},
      pages = {5–67},
      zblnumber = {0097.35901},
      url = {http://www.numdam.org/item?id=PMIHES_1960__7__5_0 },
      }
  • [St] Go to document Stack project authors, Stack project.
    @MISC{St, url = {http://stacks.math.columbia.edu},
      title = {Stack project authors, {S}tack project},
      key = {St},
      }
  • [V] 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{V, mrkey = {2265677},
      number = {3},
      issn = {0012-9593},
      author = {Viehmann, Eva},
      mrclass = {14L15 (14M15 20G25)},
      doi = {10.1016/j.ansens.2006.04.001},
      journal = {Ann. Sci. École Norm. Sup.},
      zblnumber = {1108.14036},
      volume = {39},
      mrnumber = {2265677},
      fjournal = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      mrreviewer = {Alan Koch},
      coden = {ASENAH},
      title = {The dimension of some affine {D}eligne-{L}usztig varieties},
      year = {2006},
      pages = {513--526},
      }
  • [Vi1] Go to document E. Viehmann, "Moduli spaces of $p$-divisible groups," J. Algebraic Geom., vol. 17, iss. 2, pp. 341-374, 2008.
    @ARTICLE{Vi1, mrkey = {2369090},
      number = {2},
      issn = {1056-3911},
      author = {Viehmann, Eva},
      mrclass = {14L05 (14D20)},
      doi = {10.1090/S1056-3911-07-00480-8},
      journal = {J. Algebraic Geom.},
      zblnumber = {1144.14040},
      volume = {17},
      mrnumber = {2369090},
      fjournal = {Journal of Algebraic Geometry},
      mrreviewer = {Noriko Yui},
      title = {Moduli spaces of {$p$}-divisible groups},
      year = {2008},
      pages = {341--374},
      }
  • [Vi2] Go to document E. Viehmann, "The global structure of moduli spaces of polarized $p$-divisible groups," Doc. Math., vol. 13, pp. 825-852, 2008.
    @ARTICLE{Vi2, mrkey = {2466182},
      issn = {1431-0635},
      author = {Viehmann, Eva},
      mrclass = {14L05 (14G35 14K10)},
      journal = {Doc. Math.},
      zblnumber = {1162.14033},
      volume = {13},
      mrnumber = {2466182},
      fjournal = {Documenta Mathematica},
      mrreviewer = {Alan Koch},
      title = {The global structure of moduli spaces of polarized {$p$}-divisible groups},
      year = {2008},
      pages = {825--852},
      url = {http://www.math.uiuc.edu/documenta/vol-13/24.pdf},
      }
  • [XZ] L. Xiao and X. Zhu, Cycles on Shimura varieties via geometric Satake.
    @MISC{XZ,
      author = {Xiao, L. and Zhu, X.},
      note = {in preparation},
      title = {Cycles on {S}himura varieties via geometric {S}atake},
      }
  • [Z11] X. Zhu, "The geometric Satake correspondence for ramified groups," Ann. Sci. Éc. Norm. Supér., vol. 48, iss. 2, pp. 409-451, 2015.
    @ARTICLE{Z11, mrkey = {3346175},
      number = {2},
      issn = {0012-9593},
      author = {Zhu, Xinwen},
      mrclass = {20G25 (14G35)},
      journal = {Ann. Sci. Éc. Norm. Supér.},
      zblnumber = {06456002},
      volume = {48},
      mrnumber = {3346175},
      fjournal = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      mrreviewer = {David-Alexandre Guiraud},
      title = {The geometric {S}atake correspondence for ramified groups},
      year = {2015},
      pages = {409--451},
      }
  • [Z16] X. Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence, 2016.
    @MISC{Z16,
      author = {Zhu, Xinwen},
      arxiv = {1603.05593},
      title = {An introduction to affine {G}rassmannians and the geometric {S}atake equivalence},
      year = {2016},
      }

Authors

Xinwen Zhu

California Institute of Technology, Pasadena, CA