Abstract
For a reductive group over an algebraically closed field of characteristic
Funding statement: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1144152.
Acknowledgements
I would first like to thank my advisor Mark Kisin for suggesting a topic that led to this paper and for his consistent guidance and encouragement. I would also like to express my gratitude to CĂŠdric PĂŠpin, Timo Richarz, Tobias Schmidt, and Xinwen Zhu for their comments and insights on an earlier version of this paper. I thank the referee for their careful reading of the paper and several helpful suggestions and corrections. Finally, it is a pleasure to thank Bhargav Bhatt, Justin Campbell, Dennis Gaitsgory, Michel Gros, Michael Harris, Florian Herzig, Koji Shimizu, Karen Smith, David Yang, Zijian Yao, and Yifei Zhao for their interest and helpful conversations.
References
[1] H.âH. Andersen, Schubert varieties and Demazureâs character formula, Invent. Math. 79 (1985), no. 3, 611â618. 10.1007/BF01388527Suche in Google Scholar
[2] A. Beauville and Y. Laszlo, Un lemme de descente, C. R. Acad. Sci. Paris SĂŠr. I Math. 320 (1995), no. 3, 335â340. Suche in Google Scholar
[3] A.âA. BeÄlinson, J. Bernstein and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy 1981), AstĂŠrisque 100, SociĂŠtĂŠ MathĂŠmatique de France, Paris (1982), 5â171. Suche in Google Scholar
[4] R. Bezrukavnikov, On two geometric realizations of an affine Hecke algebra, Publ. Math. Inst. Hautes Ătudes Sci. 123 (2016), 1â67. 10.1007/s10240-015-0077-xSuche in Google Scholar
[5] M.âP. Brodmann and R.âY. Sharp, Local cohomology. An algebraic introduction with geometric applications, 2nd ed., Cambridge Stud. Adv. Math. 136, Cambridge University Press, Cambridge 2013 Suche in Google Scholar
[6]
R. Cass,
Central elements in affine mod p Hecke algebras via perverse
[7] P. Deligne and J.âS. Milne, Tannakian categories, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math. 900, Springer, Berlin (1982), 101â228. 10.1007/978-3-540-38955-2_4Suche in Google Scholar
[8] M. Emerton and M. Kisin, An introduction to the RiemannâHilbert correspondence for unit F-crystals, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin (2004), 677â700. 10.1515/9783110198133.2.677Suche in Google Scholar
[9] M. Emerton and M. Kisin, The RiemannâHilbert correspondence for unit F-crystals, AstĂŠrisque 293 (2004). 10.1515/9783110198133.2.677Suche in Google Scholar
[10] G. Faltings, Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. (JEMS) 5 (2003), no. 1, 41â68. 10.1007/s10097-002-0045-xSuche in Google Scholar
[11] E. Freitag and R. Kiehl, Ătale cohomology and the Weil conjecture, Ergeb. Math. Grenzgeb. (3) 13, Springer, Berlin 1988, Translated from the German by Betty S. Waterhouse and William C. Waterhouse, With an historical introduction by J.âA. DieudonnĂŠ. 10.1007/978-3-662-02541-3Suche in Google Scholar
[12] O. Gabber, Notes on some t-structures, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin (2004), 711â734. 10.1515/9783110198133.2.711Suche in Google Scholar
[13] D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math. 144 (2001), no. 2, 253â280. 10.1007/s002220100122Suche in Google Scholar
[14] U. GĂśrtz, On the flatness of models of certain Shimura varieties of PEL-type, Math. Ann. 321 (2001), no. 3, 689â727. 10.1007/s002080100250Suche in Google Scholar
[15] U. GĂśrtz, Affine Springer fibers and affine DeligneâLusztig varieties, Affine flag manifolds and principal bundles, Trends Math., Birkhäuser/Springer, Basel (2010), 1â50. 10.1007/978-3-0346-0288-4_1Suche in Google Scholar
[16]
E. Grosse-KlĂśnne,
From pro-p Iwahori-Hecke modules to
[17] A. Grothendieck, RevĂŞtements ĂŠtales et groupe fondamental, Lecture Notes in Math. 224, Springer, Berlin 1971. 10.1007/BFb0058656Suche in Google Scholar
[18] A. Grothendieck and J. DieudonnĂŠ, ĂlĂŠments de gĂŠomĂŠtrie algĂŠbrique. IV. Ătude locale des schĂŠmas et des morphismes de schĂŠmas. IV, Publ. Math. Inst. Hautes Ătudes Sci. 32 (1967), 361â361. 10.1007/BF02732123Suche in Google Scholar
[19] P. Hamacher and E. Viehmann, Irreducible components of minuscule affine DeligneâLusztig varieties, Algebra Number Theory 12 (2018), no. 7, 1611â1634. 10.2140/ant.2018.12.1611Suche in Google Scholar
[20] M. Hashimoto, Surjectivity of multiplication and F-regularity of multigraded rings, Commutative algebra (Grenoble/Lyon 2001), Contemp. Math. 331, American Mathematical Society, Providence (2003), 153â170. 10.1090/conm/331/05908Suche in Google Scholar
[21] G. Henniart and M.-F. VignĂŠras, A Satake isomorphism for representations modulo p of reductive groups over local fields, J. Reine Angew. Math. 701 (2015), 33â75. 10.1515/crelle-2013-0021Suche in Google Scholar
[22] F. Herzig, A Satake isomorphism in characteristic p, Compos. Math. 147 (2011), no. 1, 263â283. 10.1112/S0010437X10004951Suche in Google Scholar
[23]
F. Herzig,
The classification of irreducible admissible mod p representations of a p-adic
[24] M. Hochster and C. Huneke, Tight closure and strong F-regularity, MĂŠm. Soc. Math. France (N.S.) 38 (1989), 119â133. 10.24033/msmf.343Suche in Google Scholar
[25] M. Hochster and C. Huneke, F-regularity, test elements, and smooth base change, Trans. American Mathematical Society 346 (1994), no. 1, 1â62. 10.1090/S0002-9947-1994-1273534-XSuche in Google Scholar
[26] J.âC. Jantzen, Nilpotent orbits in representation theory, Lie theory, Progr. Math. 228, Birkhäuser, Boston (2004), 1â211. 10.1007/978-0-8176-8192-0_1Suche in Google Scholar
[27] R. Kiehl and R. Weissauer, Weil conjectures, perverse sheaves and lâadic Fourier transform, Ergeb. Math. Grenzgeb. (3) 42, Springer, Berlin 2001. 10.1007/978-3-662-04576-3Suche in Google Scholar
[28] S.âJ. KovĂĄcs, Rational singularities, preprint (2017), https://arxiv.org/abs/1703.02269v7. Suche in Google Scholar
[29] V. Lafforgue, Chtoucas pour les groupes rĂŠductifs et paramĂŠtrisation de Langlands globale, J. Amer. Math. Soc. 31 (2018), no. 3, 719â891. 10.1090/jams/897Suche in Google Scholar
[30]
N. Lauritzen, U. Raben-Pedersen and J.âF. Thomsen,
Global F-regularity of Schubert varieties with applications to
[31] N. Lauritzen and J.âF. Thomsen, Line bundles on BottâSamelson varieties, J. Algebraic Geom. 13 (2004), no. 3, 461â473. 10.1090/S1056-3911-03-00358-8Suche in Google Scholar
[32] E. Letellier, Fourier transforms of invariant functions on finite reductive Lie algebras, Lecture Notes in Math. 1859, Springer, Berlin 2005. 10.1007/b104209Suche in Google Scholar
[33] J. Lipman and B. Teissier, Pseudorational local rings and a theorem of BriançonâSkoda about integral closures of ideals, Michigan Math. J. 28 (1981), no. 1, 97â116. 10.1307/mmj/1029002461Suche in Google Scholar
[34] O. Mathieu, Formules de caractères pour les algèbres de KacâMoody gĂŠnĂŠrales, AstĂŠrisque 159â160 (1988), 267â267. Suche in Google Scholar
[35] V.âB. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2) 122 (1985), no. 1, 27â40. 10.2307/1971368Suche in Google Scholar
[36] V.âB. Mehta and V. Srinivas, Normality of Schubert varieties, Amer. J. Math. 109 (1987), no. 5, 987â989. 10.2307/2374497Suche in Google Scholar
[37] J.âS. Milne and K. Shih, Conjugates of Shimura varieties, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math. 900, Springer, Berlin (1982), 280â356. 10.1007/978-3-540-38955-2_7Suche in Google Scholar
[38] I. MirkoviÄ and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), no. 1, 95â143. 10.4007/annals.2007.166.95Suche in Google Scholar
[39] B.âC. NgĂ´ and P. Polo, RĂŠsolutions de Demazure affines et formule de CasselmanâShalika gĂŠomĂŠtrique, J. Algebraic Geom. 10 (2001), no. 3, 515â547. Suche in Google Scholar
[40] R. Ollivier, Compatibility between Satake and Bernstein isomorphisms in characteristic p, Algebra Number Theory 8 (2014), no. 5, 1071â1111. 10.2140/ant.2014.8.1071Suche in Google Scholar
[41] G. Pappas and M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), no. 1, 118â198, With an appendix by T. Haines and Rapoport. 10.1016/j.aim.2008.04.006Suche in Google Scholar
[42]
C. PĂŠpin and T. Schmidt,
Generic and mod p KazhdanâLusztig theory for
[43] S. Ramanan and A. Ramanathan, Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), no. 2, 217â224. 10.1007/BF01388970Suche in Google Scholar
[44] T. Richarz, A new approach to the geometric Satake equivalence, Doc. Math. 19 (2014), 209â246. 10.4171/dm/445Suche in Google Scholar
[45] T. Richarz, Affine Grassmannians and geometric Satake equivalences, Int. Math. Res. Not. IMRN 2016 (2016), no. 12, 3717â3767. 10.1093/imrn/rnv226Suche in Google Scholar
[46] C.âS. Seshadri, Line bundles on Schubert varieties, Vector bundles on algebraic varieties (Bombay 1984), Tata Inst. Fund. Res. Stud. Math. 11, Tata Inst. Fund. Res., Bombay (1987), 499â528. Suche in Google Scholar
[47] K.âE. Smith, F-rational rings have rational singularities, Amer. J. Math. 119 (1997), no. 1, 159â180. 10.1353/ajm.1997.0007Suche in Google Scholar
[48] K.âE. Smith, Globally F-regular varieties: Applications to vanishing theorems for quotients of Fano varieties, Michigan Math. J. 48 (2000), 553â572. 10.1307/mmj/1030132733Suche in Google Scholar
[49]
A. Stäbler,
Intermediate extensions of perverse constructible
[50] J.âD. VĂŠlez, Openness of the F-rational locus and smooth base change, J. Algebra 172 (1995), no. 2, 425â453. 10.1016/S0021-8693(05)80010-9Suche in Google Scholar
[51]
M.-F. VignĂŠras,
Pro-p-Iwahori Hecke ring and supersingular
[52] X. Zhu, Affine Demazure modules and T-fixed point subschemes in the affine Grassmannian, Adv. Math. 221 (2009), no. 2, 570â600. 10.1016/j.aim.2009.01.003Suche in Google Scholar
[53] X. Zhu, On the coherence conjecture of Pappas and Rapoport, Ann. of Math. (2) 180 (2014), no. 1, 1â85. 10.4007/annals.2014.180.1.1Suche in Google Scholar
[54] X. Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence, Geometry of moduli spaces and representation theory, IAS/Park City Math. Ser. 24, American Mathematical Society, Providence (2017), 59â154. 10.1090/pcms/024/02Suche in Google Scholar
[55] The Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu, 2021. Suche in Google Scholar
Š 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- A central limit theorem for integrals of random waves
- Connectedness of Kisin varieties associated to absolutely irreducible Galois representations
- The HilbertâSchinzel specialization property
- A mixed elliptic-parabolic boundary value problem coupling a harmonic-like map with a nonlinear spinor
- Isomorphisms among quantum Grothendieck rings and propagation of positivity
- The NeukirchâUchida theorem with restricted ramification
- Perverse đ˝p-sheaves on the affine Grassmannian
- Corrigendum to Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space (J. reine angew. Math. 727 (2017), 269â299)
Artikel in diesem Heft
- Frontmatter
- A central limit theorem for integrals of random waves
- Connectedness of Kisin varieties associated to absolutely irreducible Galois representations
- The HilbertâSchinzel specialization property
- A mixed elliptic-parabolic boundary value problem coupling a harmonic-like map with a nonlinear spinor
- Isomorphisms among quantum Grothendieck rings and propagation of positivity
- The NeukirchâUchida theorem with restricted ramification
- Perverse đ˝p-sheaves on the affine Grassmannian
- Corrigendum to Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space (J. reine angew. Math. 727 (2017), 269â299)