Local models in the ramified case. I: The EL-case
Authors:
G. Pappas and M. Rapoport
Journal:
J. Algebraic Geom. 12 (2003), 107-145
DOI:
https://doi.org/10.1090/S1056-3911-02-00334-X
Published electronically:
July 18, 2002
MathSciNet review:
1948687
Full-text PDF
Abstract |
References |
Additional Information
Abstract: Local models are schemes defined in linear algebra terms that describe the étale local structure of integral models for Shimura varieties and other moduli spaces. We point out that the flatness conjecture of Rapoport-Zink on local models fails in the presence of ramification and that in that case one has to modify their definition. We study modifications of the local models for $G=\text \textrm {R}_{E/F}\text \textrm {GL}(n)$, with $E/F$ a totally ramified extension, and for a maximal parahoric level subgroup. The special fibers of these models are subschemes of the affine Grassmannian. We give applications to the structure of Schubert varieties in the affine Grassmannian and to the calculation of sheaves of nearby cycles and describe a relation with geometric convolution. In the general EL case, we replace the conjecture of Rapoport-Zink with a conjecture about the modified local models.
[B-L]B-L A. Beauville–Y. Laszlo: Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), 385–419.
[BBD]BBD A. Beilinson–J. Bernstein–P. Deligne: Faisceaux pervers, Astérisque 100 (1982), 3–171.
[BM]BM W. Borho–R. MacPherson: Partial resolutions of nilpotent varieties, Astérisque 101-102 (1983), 23-74.
[BG]BG A. Braverman–D. Gaitsgory: On Ginzburg’s Lagrangian construction of representations of $GL_n$, Math. Res. Lett. 6 (1999), 195–201.
[deC-P]deC-P C. De Concini–C. Procesi: Symmetric Functions, Conjugacy Classes and the Flag Variety, Invent. Math. 64 (1981), 203–219.
[D]D P. Deligne: Théorèmes de finitude en cohomologie $\ell$-adique, in SGA $4\frac {1}{2}$, 233–261, SLN 569, Springer-Verlag 1977.
[D-P]D-P P. Deligne–G. Pappas: Singularités des espaces de modules de Hilbert, en les caractéristiques divisants le discriminant, Compositio Math. 90 (1994), 59–79.
[E-S]E-S D. Eisenbud–D. Saltman: Rank varieties of matrices, Commutative Algebra, MSRI Publications 15 (1989), Hochster, Huneke, Sally ed., Springer Verlag.
[F]F G. Faltings: Algebraic loop groups and moduli spaces of bundles, preprint Max-Planck-Institut, Bonn 2000, 29 p.
[FGKV]FGKV E. Frenkel–D. Gaitsgory–D. Kazhdan–K. Vilonen: Geometric realization of Whittaker functions and the Langlands conjecture, J. Amer. Math. Soc. 11 (1998), 451–484.
[Gi]Gi V. Ginzburg: Perverse sheaves on a loop group and Langlands duality, alg-geom/9511007.
[G]G U. Görtz: On the flatness of models of certain Shimura varieties of PEL type, Math. Ann. 321 (2001), 689–727.
[G1]G1 U. Görtz: On the flatness of local models for the symplectic group, preprint Köln 2000.
[HN]HN T. Haines–B. C. Ngô: Nearby cycles for local models of some Shimura varieties, preprint 1999.
[HS]HS R. Hotta–T. Springer: A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups, Invent. Math. 41 (1977), 113–127.
[I]I L. Illusie: Autour du théorème de monodromie locale, Astérisque 223 (1994), 9–58.
[K]K R. Kottwitz: Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373–444.
[KR]KR R. Kottwitz–M. Rapoport: Minuscule alcoves for $GL_n$ and $GSp_{2n}$, Manuscripta Math. 102 (2000), 403–428.
[LMoB]LMoB G. Laumon–L. Moret-Bailly. Champs algébriques. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Vol. 39. Springer–Verlag Berlin Heidelberg 2000.
[Li]Li P. Littelmann: Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras. Journ. AMS 11 3 (1999) pp. 551–567.
[L]L G. Lusztig: Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), 169–178.
[Ma]Ma H. Matsumura: Commutative Algebra. Benjamin/Cummings Publishing Co., Reading, Mass. 1980.
[Mat]Mat O. Mathieu: Formules de caractères pour les algèbres de Kac-Moody générales, Astérisque 159-160 (1988).
[M-vdK]M-vdK V.B. Mehta–W. van der Kallen: A simultaneous Frobenius splitting for closures of conjugacy classes of nilpotent matrices, Compositio Math. 84 (1992), 211–221.
[Mc]Mc I. Macdonald: Symmetric functions and Hall polynomials, $2^{\text \textrm {nd}}$ ed., Oxford 1995.
[M-V]M-V I. Mirkovic–K. Vilonen: Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett. 7 (2000), 13–24.
[N-P]N-P B.C. Ngô–P. Polo: Résolutions de Demazure affines et formule de Casselman-Shalika géométrique, J. Alg. Geom. 10 (2001), 514–547.
[P]P G. Pappas: Local structure of arithmetic moduli for PEL Shimura varieties, J. Alg. Geom. 9 (2000), 577–605.
[RZ]RZ M. Rapoport–Th. Zink: Period spaces for $p$-divisible groups. Ann. of Math. Studies, vol. 141, Princeton University Press 1996.
[Sp]Sp N. Spaltenstein: The fixed point set of a unipotent transformation on the flag manifold, Proc. Kon. Ak. v. Wet. 79 (5) (1976), 452–456.
[St]St E. Strickland: On the variety of projectors, J. Algebra 106 (1987), 135–147.
[W]W J. Weyman: Two results on equations of nilpotent orbits, J. Alg. Geom., to appear.
[B-L]B-L A. Beauville–Y. Laszlo: Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), 385–419.
[BBD]BBD A. Beilinson–J. Bernstein–P. Deligne: Faisceaux pervers, Astérisque 100 (1982), 3–171.
[BM]BM W. Borho–R. MacPherson: Partial resolutions of nilpotent varieties, Astérisque 101-102 (1983), 23-74.
[BG]BG A. Braverman–D. Gaitsgory: On Ginzburg’s Lagrangian construction of representations of $GL_n$, Math. Res. Lett. 6 (1999), 195–201.
[deC-P]deC-P C. De Concini–C. Procesi: Symmetric Functions, Conjugacy Classes and the Flag Variety, Invent. Math. 64 (1981), 203–219.
[D]D P. Deligne: Théorèmes de finitude en cohomologie $\ell$-adique, in SGA $4\frac {1}{2}$, 233–261, SLN 569, Springer-Verlag 1977.
[D-P]D-P P. Deligne–G. Pappas: Singularités des espaces de modules de Hilbert, en les caractéristiques divisants le discriminant, Compositio Math. 90 (1994), 59–79.
[E-S]E-S D. Eisenbud–D. Saltman: Rank varieties of matrices, Commutative Algebra, MSRI Publications 15 (1989), Hochster, Huneke, Sally ed., Springer Verlag.
[F]F G. Faltings: Algebraic loop groups and moduli spaces of bundles, preprint Max-Planck-Institut, Bonn 2000, 29 p.
[FGKV]FGKV E. Frenkel–D. Gaitsgory–D. Kazhdan–K. Vilonen: Geometric realization of Whittaker functions and the Langlands conjecture, J. Amer. Math. Soc. 11 (1998), 451–484.
[Gi]Gi V. Ginzburg: Perverse sheaves on a loop group and Langlands duality, alg-geom/9511007.
[G]G U. Görtz: On the flatness of models of certain Shimura varieties of PEL type, Math. Ann. 321 (2001), 689–727.
[G1]G1 U. Görtz: On the flatness of local models for the symplectic group, preprint Köln 2000.
[HN]HN T. Haines–B. C. Ngô: Nearby cycles for local models of some Shimura varieties, preprint 1999.
[HS]HS R. Hotta–T. Springer: A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups, Invent. Math. 41 (1977), 113–127.
[I]I L. Illusie: Autour du théorème de monodromie locale, Astérisque 223 (1994), 9–58.
[K]K R. Kottwitz: Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373–444.
[KR]KR R. Kottwitz–M. Rapoport: Minuscule alcoves for $GL_n$ and $GSp_{2n}$, Manuscripta Math. 102 (2000), 403–428.
[LMoB]LMoB G. Laumon–L. Moret-Bailly. Champs algébriques. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Vol. 39. Springer–Verlag Berlin Heidelberg 2000.
[Li]Li P. Littelmann: Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras. Journ. AMS 11 3 (1999) pp. 551–567.
[L]L G. Lusztig: Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), 169–178.
[Ma]Ma H. Matsumura: Commutative Algebra. Benjamin/Cummings Publishing Co., Reading, Mass. 1980.
[Mat]Mat O. Mathieu: Formules de caractères pour les algèbres de Kac-Moody générales, Astérisque 159-160 (1988).
[M-vdK]M-vdK V.B. Mehta–W. van der Kallen: A simultaneous Frobenius splitting for closures of conjugacy classes of nilpotent matrices, Compositio Math. 84 (1992), 211–221.
[Mc]Mc I. Macdonald: Symmetric functions and Hall polynomials, $2^{\text \textrm {nd}}$ ed., Oxford 1995.
[M-V]M-V I. Mirkovic–K. Vilonen: Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett. 7 (2000), 13–24.
[N-P]N-P B.C. Ngô–P. Polo: Résolutions de Demazure affines et formule de Casselman-Shalika géométrique, J. Alg. Geom. 10 (2001), 514–547.
[P]P G. Pappas: Local structure of arithmetic moduli for PEL Shimura varieties, J. Alg. Geom. 9 (2000), 577–605.
[RZ]RZ M. Rapoport–Th. Zink: Period spaces for $p$-divisible groups. Ann. of Math. Studies, vol. 141, Princeton University Press 1996.
[Sp]Sp N. Spaltenstein: The fixed point set of a unipotent transformation on the flag manifold, Proc. Kon. Ak. v. Wet. 79 (5) (1976), 452–456.
[St]St E. Strickland: On the variety of projectors, J. Algebra 106 (1987), 135–147.
[W]W J. Weyman: Two results on equations of nilpotent orbits, J. Alg. Geom., to appear.
Additional Information
G. Pappas
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email:
pappas@math.msu.edu
M. Rapoport
Affiliation:
Mathematisches Institut der Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany
MR Author ID:
144965
Email:
rapoport@mi.uni-koeln.de
Received by editor(s):
August 30, 2000
Received by editor(s) in revised form:
November 27, 2001
Published electronically:
July 18, 2002
Additional Notes:
The first author was partially supported by NSF grant DMS99-70378 and by a Sloan Research Fellowship
