On a conjecture of Kottwitz and Rapoport  [ Une conjecture de Kottwitz et Rapoport ]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 6, pp. 1017-1038.

On démontre une conjecture de Kottwitz et Rapoport sur une réciproque à l'inégalité de Mazur pour tout groupe (connexe) réductif, déployé ou quasi-déployé non-ramifié. Nos résultats sont liés à la non-vacuité de certaines variétés de Deligne-Lusztig affines.

We prove a conjecture of Kottwitz and Rapoport which implies a converse to Mazur's Inequality for all (connected) split and quasi-split unramified reductive groups. Our results are related to the non-emptiness of certain affine Deligne-Lusztig varieties.

DOI : https://doi.org/10.24033/asens.2138
Classification : 14L15,  14M15,  20G25
Mots clés : polygone de Newton, isocristal, variétés de Deligne-Lusztig affines
@article{ASENS_2010_4_43_6_1017_0,
     author = {Gashi, Q\"endrim R.},
     title = {On a conjecture of Kottwitz and Rapoport},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1017--1038},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 43},
     number = {6},
     year = {2010},
     doi = {10.24033/asens.2138},
     zbl = {1225.14037},
     mrnumber = {2778454},
     language = {en},
     url = {www.numdam.org/item/ASENS_2010_4_43_6_1017_0/}
}
Gashi, Qëndrim R. On a conjecture of Kottwitz and Rapoport. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 6, pp. 1017-1038. doi : 10.24033/asens.2138. http://www.numdam.org/item/ASENS_2010_4_43_6_1017_0/

[1] J. Arthur, A local trace formula, Publ. Math. I.H.É.S. 73 (1991), 5-96. | Numdam | MR 1114210 | Zbl 0741.22013

[2] N. Bourbaki, Lie groups and Lie algebras. Chapters 4-6, Elements of Mathematics, Springer, 2002. | MR 1890629 | Zbl 0983.17001

[3] N. Bourbaki, Lie groups and Lie algebras, Chapters 7-9, Elements of Mathematics, Springer, 2005. | MR 2109105 | Zbl 1139.17002

[4] Q. R. Gashi, A vanishing result for toric varieties associated with root systems, Albanian J. Math. 1 (2007), 235-244. | MR 2367216 | Zbl 1133.14054

[5] Q. R. Gashi, Vanishing results for toric varieties associated to GL n and G 2 , Transform. Groups 13 (2008), 149-171. | MR 2421320 | Zbl 1160.14039

[6] Q. R. Gashi & T. Schedler, On dominance and minuscule Weyl group elements, preprint arXiv:0908.1091. | MR 2772538 | Zbl 1239.20045

[7] M. Görtz, T. J. Haines, R. E. Kottwitz & D. C. Reuman, Affine Deligne-Lusztig varieties in affine flag varieties, preprint arXiv:0805.0045. | Zbl 1229.14036

[8] U. Görtz, T. J. Haines, R. E. Kottwitz & D. C. Reuman, Dimensions of some affine Deligne-Lusztig varieties, Ann. Sci. École Norm. Sup. 39 (2006), 467-511. | Numdam | MR 2265676 | Zbl 1108.14035

[9] R. E. Kottwitz, Isocrystals with additional structure, Compositio Math. 56 (1985), 201-220. | Numdam | MR 809866 | Zbl 0597.20038

[10] R. E. Kottwitz, On the Hodge-Newton decomposition for split groups, Int. Math. Res. Not. 2003 (2003), 1433-1447. | MR 1976046 | Zbl 1074.14016

[11] R. E. Kottwitz & M. Rapoport, On the existence of F-crystals, Comment. Math. Helv. 78 (2003), 153-184. | MR 1966756 | Zbl 1126.14023

[12] C. Lucarelli, A converse to Mazur's inequality for split classical groups, J. Inst. Math. Jussieu 3 (2004), 165-183. | MR 2055708 | Zbl 1054.14059

[13] C. Lucarelli, A converse to Mazur's inequality for split classical groups, Thèse, University of Chicago, 2004. | MR 2055708 | Zbl 1054.14059

[14] B. Mazur, Frobenius and the Hodge filtration, Bull. Amer. Math. Soc. 78 (1972), 653-667. | MR 330169 | Zbl 0258.14006

[15] B. Mazur, Frobenius and the Hodge filtration (estimates), Ann. of Math. 98 (1973), 58-95. | MR 321932 | Zbl 0261.14005

[16] S. Mozes, Reflection processes on graphs and Weyl groups, J. Combin. Theory Ser. A 53 (1990), 128-142. | MR 1031617 | Zbl 0741.05035

[17] M. Rapoport, A positivity property of the Satake isomorphism, Manuscripta Math. 101 (2000), 153-166. | MR 1742251 | Zbl 0941.22006

[18] M. Rapoport, A guide to the reduction modulo p of Shimura varieties, Astérisque 298 (2005), 271-318. | MR 2141705 | Zbl 1084.11029

[19] M. Rapoport & M. Richartz, On the classification and specialization of F-isocrystals with additional structure, Compositio Math. 103 (1996), 153-181. | Numdam | MR 1411570 | Zbl 0874.14008

[20] T. A. Springer, Linear algebraic groups, 2nd éd., Progress in Mathematics, Birkhäuser, 2006. | MR 632835 | Zbl 0927.20024

[21] J. R. Stembridge, The partial order of dominant weights, Adv. Math. 136 (1998), 340-364. | MR 1626860 | Zbl 0916.06001

[22] J. R. Stembridge, Minuscule elements of Weyl groups, J. Algebra 235 (2001), 722-743. | MR 1805477 | Zbl 0973.17034

[23] E. Viehmann, The dimension of some affine Deligne-Lusztig varieties, Ann. Sci. École Norm. Sup. 39 (2006), 513-526. | Numdam | MR 2265677 | Zbl 1108.14036

[24] J.-P. Wintenberger, Existence de F-cristaux avec structures supplémentaires, Adv. Math. 190 (2005), 196-224. | MR 2104909 | Zbl 1104.14013