×

On the coherence conjecture of Pappas and Rapoport. (English) Zbl 1300.14042

Local models were introduced by M. Rapoport and Th. Zink [Period spaces for \(p\)-divisible groups. Annals of Mathematics Studies. 141. Princeton, NJ: Princeton Univ. Press. (1996; Zbl 0873.14039)] in order to study the PEL-type Shimura varieties. The so-called “naive” local model, which is a scheme over the ring of integers of the reflex field, is unfortunately not flat in general. To remedy this defect, a common strategy is to take the flat closure of the generic fiber; however, the moduli interpretation is lost in this procedure.
It turns out that questions about the special fibers of these flat models become tractable if one admits the Coherence Conjecture of Pappas and Rapoport, which was motivated by the case of ramified unitary groups. To state this conjecture, take \(k\) to be an algebraically closed field and set \(F = k((t))\), \(\mathcal{O} = k[[t]]\). Let \(G\) be a connected reductive \(F\)-group that splits over a tamely ramified extension \(\tilde{F}/F\). Choose a maximal torus \(T\) containing a maximal split torus.
To define local models, we consider a geometric conjugacy class of homomorphisms \(\mu: \mathbb{G}_{m, \tilde{F}} \to G \times_F \tilde{F}\). If \(\mu\) is a minuscule coweight, we may define the corresponding partial flag variety \(X(\mu) = H/P(\mu)\) where \(H\) is the Chevalley group attached to \(G\). Let \(\mathcal{L}(\mu)\) stand for the ample generator of \(\mathrm{Pic} X(\mu)\) and put \(h_\mu(a) = \dim_k \Gamma(X(\mu), \mathcal{L}(\mu)^a)\). In general, if \(\mu = \mu_1 + \cdots + \mu_r\) is a sum of minuscule coweights, we put \(h_\mu = h_{\mu_1} \cdots h_{\mu_r}\).
On the other hand, let \(Y\) be a subset of a chosen set of simple affine roots of \((G, T, \cdots)\). One may define the \(k\)-variety \(\mathcal{A}^Y(\mu)^\circ\) which is a union of certain Schubert varieties for the loop group \(LG_{\mathrm{sc}}\) indexed by the admissible set \(\mathrm{Adm}^Y(\mu)\). The set \(Y\) also gives rise to the Schubert variety \(\mathrm{Fl}^Y := LG/L^+ G_{\sigma_Y}\), with \(\sigma_Y\) being the corresponding facet in the Bruhat-Tits building. Let \(\mathcal{L}\) be an ample line bundle on \(\mathrm{Fl}^Y\). In its modified form stated in this paper, the Coherence Conjecture asserts that \[ \dim_k \Gamma(\mathcal{A}^Y(\mu)^{\circ}, \mathcal{L}^a) = h_\mu(c(\mathcal{L})a) \] where \(c(\mathcal{L})\) is the central charge of \(\mathcal{L}\) (think in terms of Kac-Moody groups).
The proof is too involved to be summarized here. One important ingredient is the construction of the global Grassmannian \(\mathrm{Gr}_{\mathcal{G}}\) and the “global Schubert variety” \(\overline{\mathrm{Gr}_{\mathcal{G}, \mu}}\), over a suitable curve \(C\) and its ramified cover \(\tilde{C} \to C\). A crucial step is to show that the central charge of a line bundle on \(\mathrm{Gr}_{\mathcal{G}}\) is constant along the curve \(C(k)\).
This paper corrects several mistakes in [G. Pappas and M. Rapoport, Adv. Math. 219, No. 1, 118–198 (2008; Zbl 1159.22010)] and [J. Heinloth, Math. Ann. 347, No. 3, 499–528 (2010; Zbl 1193.14014)].

MSC:

14L05 Formal groups, \(p\)-divisible groups
14M15 Grassmannians, Schubert varieties, flag manifolds
14G20 Local ground fields in algebraic geometry
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] S. Arkhipov and R. Bezrukavnikov, ”Perverse sheaves on affine flags and Langlands dual group,” Israel J. Math., vol. 170, pp. 135-183, 2009. · Zbl 1214.14011 · doi:10.1007/s11856-009-0024-y
[2] A. Beauville and Y. Laszlo, ”Conformal blocks and generalized theta functions,” Comm. Math. Phys., vol. 164, iss. 2, pp. 385-419, 1994. · Zbl 0815.14015 · doi:10.1007/BF02101707
[3] A. Beauville and Y. Laszlo, ”Un lemme de descente,” C. R. Acad. Sci. Paris Sér. I Math., vol. 320, iss. 3, pp. 335-340, 1995. · Zbl 0852.13005
[4] A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves. · Zbl 0864.14007
[5] R. Bezrukavnikov, ”On tensor categories attached to cells in affine Weyl groups,” in Representation Theory of Algebraic Groups and Quantum Groups, Tokyo: Math. Soc. Japan, 2004, vol. 40, pp. 69-90. · Zbl 1078.20044
[6] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, New York: Springer-Verlag, 1990, vol. 21. · Zbl 0705.14001 · doi:10.1007/978-3-642-51438-8
[7] M. Brion and S. Kumar, Frobenius Splitting Methods in Geometry and Representation Theory, Boston, MA: Birkhäuser, 2005, vol. 231. · Zbl 1072.14066
[8] F. Bruhat and J. Tits, ”Groupes réductifs sur un corps local,” Inst. Hautes Études Sci. Publ. Math., vol. 41, pp. 5-251, 1972. · Zbl 0254.14017 · doi:10.1007/BF02715544
[9] 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. · Zbl 0597.14041 · doi:10.1007/BF02700560
[10] C. Chai and P. Norman, ”Singularities of the \(\Gamma_0(p)\)-level structure,” J. Algebraic Geom., vol. 1, iss. 2, pp. 251-278, 1992. · Zbl 0785.14001
[11] B. Conrad, Reductive group schemes. · Zbl 1349.14151
[12] A. J. de Jong, ”The moduli spaces of principally polarized abelian varieties with \(\Gamma_0(p)\)-level structure,” J. Algebraic Geom., vol. 2, iss. 4, pp. 667-688, 1993. · Zbl 0816.14020
[13] P. Deligne and G. Pappas, ”Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant,” Compositio Math., vol. 90, iss. 1, pp. 59-79, 1994. · Zbl 0826.14027
[14] B. Edixhoven, ”Néron models and tame ramification,” Compositio Math., vol. 81, iss. 3, pp. 291-306, 1992. · Zbl 0759.14033
[15] 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, p. 231, 1965. · Zbl 0135.39701
[16] G. Faltings, ”Algebraic loop groups and moduli spaces of bundles,” J. Eur. Math. Soc. \((\)JEMS\()\), vol. 5, iss. 1, pp. 41-68, 2003. · Zbl 1020.14002 · doi:10.1007/s10097-002-0045-x
[17] G. Fourier and P. Littelmann, ”Tensor product structure of affine Demazure modules and limit constructions,” Nagoya Math. J., vol. 182, pp. 171-198, 2006. · Zbl 1143.22010
[18] E. Frenkel and X. Zhu, ”Gerbal representations of double loop groups,” Int. Math. Res. Not., vol. 2012, iss. 17, pp. 3929-4013, 2012. · Zbl 1280.22024 · doi:10.1093/imrn/rnr159
[19] D. Gaitsgory, ”Construction of central elements in the affine Hecke algebra via nearby cycles,” Invent. Math., vol. 144, iss. 2, pp. 253-280, 2001. · Zbl 1072.14055 · doi:10.1007/s002220100122
[20] U. Görtz, ”On the flatness of models of certain Shimura varieties of PEL-type,” Math. Ann., vol. 321, iss. 3, pp. 689-727, 2001. · Zbl 1073.14526 · doi:10.1007/s002080100250
[21] U. Görtz, ”On the flatness of local models for the symplectic group,” Adv. Math., vol. 176, iss. 1, pp. 89-115, 2003. · Zbl 1051.14027 · doi:10.1016/S0001-8708(02)00062-2
[22] T. Haines and B. C. Ngô, ”Nearby cycles for local models of some Shimura varieties,” Compositio Math., vol. 133, iss. 2, pp. 117-150, 2002. · Zbl 1009.11042 · doi:10.1023/A:1019666710051
[23] T. Haines and M. Rapoport, On paregoric subgroups. · Zbl 1159.22010 · doi:10.1016/j.aim.2008.04.006
[24] J. Heinloth, ”Uniformization of \(\mathcalG\)-bundles,” Math. Ann., vol. 347, iss. 3, pp. 499-528, 2010. · Zbl 1193.14014 · doi:10.1007/s00208-009-0443-4
[25] J. Heinloth, B. Ngô, and Z. Yun, ”Kloosterman sheaves for reductive groups,” Ann. of Math., vol. 177, iss. 1, pp. 241-310, 2013. · Zbl 1272.14012 · doi:10.4007/annals.2013.177.1.5
[26] L. Illusie, ”Autour du théorème de monodromie locale,” in Périodes \(p\)-Adiques, Paris: Soc. Math. France, 1994, vol. 223, pp. 9-57. · Zbl 0837.14013
[27] V. G. Kac, Infinite-dimensional Lie algebras, Third ed., Cambridge: Cambridge Univ. Press, 1990. · Zbl 0716.17022 · doi:10.1017/CBO9780511626234
[28] R. E. Kottwitz, ”Shimura varieties and twisted orbital integrals,” Math. Ann., vol. 269, iss. 3, pp. 287-300, 1984. · Zbl 0533.14009 · doi:10.1007/BF01450697
[29] R. E. Kottwitz, ”Isocrystals with additional structure. II,” Compositio Math., vol. 109, iss. 3, pp. 255-339, 1997. · Zbl 0533.14009 · doi:10.1007/BF01450697
[30] S. Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Boston, MA: Birkhäuser, 2002, vol. 204. · Zbl 1026.17030 · doi:10.1007/978-1-4612-0105-2
[31] O. Mathieu, ”Formules de caractères pour les algèbres de Kac-Moody générales,” Astérisque, iss. 159-160, 1988. · Zbl 0683.17010
[32] V. B. Mehta and A. Ramanathan, ”Frobenius splitting and cohomology vanishing for Schubert varieties,” Ann. of Math., vol. 122, iss. 1, pp. 27-40, 1985. · Zbl 0601.14043 · doi:10.2307/1971368
[33] 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. · Zbl 1138.22013 · doi:10.4007/annals.2007.166.95
[34] G. Pappas, ”On the arithmetic moduli schemes of PEL Shimura varieties,” J. Algebraic Geom., vol. 9, iss. 3, pp. 577-605, 2000. · Zbl 0978.14023
[35] G. Pappas and M. Rapoport, ”Local models in the ramified case. I. The EL-case,” J. Algebraic Geom., vol. 12, iss. 1, pp. 107-145, 2003. · Zbl 1063.14029 · doi:10.1090/S1056-3911-02-00334-X
[36] G. Pappas and M. Rapoport, ”Local models in the ramified case. II. Splitting models,” Duke Math. J., vol. 127, iss. 2, pp. 193-250, 2005. · Zbl 1126.14028 · doi:10.1215/S0012-7094-04-12721-6
[37] G. Pappas and M. Rapoport, ”Twisted loop groups and their affine flag varieties,” Adv. Math., vol. 219, iss. 1, pp. 118-198, 2008. · Zbl 1159.22010 · doi:10.1016/j.aim.2008.04.006
[38] G. Pappas and M. Rapoport, ”Local models in the ramified case. III. Unitary groups,” J. Inst. Math. Jussieu, vol. 8, iss. 3, pp. 507-564, 2009. · Zbl 1185.14018 · doi:10.1017/S1474748009000139
[39] G. Pappas and M. Rapoport, ”Some questions about \(\mathcalG\)-bundles on curves,” in Algebraic and Arithmetic Structures of Moduli Spaces, Tokyo: Math. Soc. Japan, 2010, vol. 58, pp. 159-171. · Zbl 1213.14028
[40] G. Pappas, M. Rapoport, and B. D. Smithling, ”Local models of Shimura varieties, I. Geometry and combinatorics,” in Handbook of Moduli: Volume III, Farkas, G. and Morrison, I., Eds., Somerville, MA: Internat. Press, 2013, vol. 26, pp. 135-217. · Zbl 1260.14003
[41] G. Pappas and X. Zhu, ”Local models of Shimura varieties and a conjecture of Kottwitz,” Invent. Math., vol. 194, iss. 1, pp. 147-254, 2013. · Zbl 1294.14012 · doi:10.1007/s00222-012-0442-z
[42] 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. · Zbl 1084.11029
[43] M. Rapoport and . T. Zink, Period spaces for \(p\)-divisible groups, Princeton, NJ: Princeton Univ. Press, 1996. · Zbl 0873.14039 · doi:10.1515/9781400882601
[44] T. Richarz, Local models and Schubert varieties in twisted affine Grassmannians. · Zbl 1315.14066 · doi:10.1016/j.jalgebra.2012.11.013
[45] R. Steinberg, ”Regular elements of semisimple algebraic groups,” Inst. Hautes Études Sci. Publ. Math., vol. 25, pp. 49-80, 1965. · Zbl 0136.30002 · doi:10.1007/BF02684397
[46] X. Zhu, ”Affine Demazure modules and \(T\)-fixed point subschemes in the affine Grassmannian,” Adv. Math., vol. 221, iss. 2, pp. 570-600, 2009. · Zbl 1167.14033 · doi:10.1016/j.aim.2009.01.003
[47] Schémas en Groupes. III: Structure des Schémas en Groupes Réductifs, Demazure, M. and Grothendieck, A., Eds., New York: Springer-Verlag, 1970, vol. 153. · Zbl 0212.52810 · doi:10.1007/BFb0059027
[48] X. Zhu, The geometric Satake correspondence for ramified groups. · Zbl 1392.11036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.