Dualities for root systems with automorphisms and applications to non-split groups
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- Represent. Theory 22 (2018), 1-26 Request permission
Abstract:
This article establishes some elementary dualities for root systems with automorphisms. We give several applications to reductive groups over non-archimedean local fields: (1) the proof of a conjecture of Pappas-Rapoport-Smithling characterizing the extremal elements of the $\{ \mu \}$-admissible sets attached to general non-split groups; (2) for quasi-split groups, a simple uniform description of the Bruhat-Tits échelonnage root system $\Sigma _0$, the Knop root system $\widetilde {\Sigma }_0$ and the Macdonald root system $\Sigma _1$, in terms of Galois actions on the absolute roots $\Phi$; and (3) for quasi-split groups, the construction of the geometric basis of the center of a parahoric Hecke algebra, and the expression of certain important elements of the stable Bernstein center in terms of this basis. The latter gives an explicit form of the test function conjecture for general Shimura varieties with parahoric level structure.References
- H. H. Andersen, Schubert varieties and Demazure’s character formula, Invent. Math. 79 (1985), no. 3, 611–618. MR 782239, DOI 10.1007/BF01388527
- F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 327923, DOI 10.1007/BF02715544
- Neil Chriss, A geometric construction of the Iwahori-Hecke algebra for unramified groups, Pacific J. Math. 179 (1997), no. 1, 11–57. MR 1452524, DOI 10.2140/pjm.1997.179.11
- D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math. 144 (2001), no. 2, 253–280. MR 1826370, DOI 10.1007/s002220100122
- Ulrich Görtz, Alcove walks and nearby cycles on affine flag manifolds, J. Algebraic Combin. 26 (2007), no. 4, 415–430. MR 2341858, DOI 10.1007/s10801-007-0063-6
- Ulrich Görtz and Thomas J. Haines, The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties, J. Reine Angew. Math. 609 (2007), 161–213. MR 2350783, DOI 10.1515/CRELLE.2007.063
- Thomas J. Haines, The combinatorics of Bernstein functions, Trans. Amer. Math. Soc. 353 (2001), no. 3, 1251–1278. MR 1804418, DOI 10.1090/S0002-9947-00-02716-1
- Thomas J. Haines, The stable Bernstein center and test functions for Shimura varieties, Automorphic forms and Galois representations. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 415, Cambridge Univ. Press, Cambridge, 2014, pp. 118–186. MR 3444233
- Thomas J. Haines, On Satake parameters for representations with parahoric fixed vectors, Int. Math. Res. Not. IMRN 20 (2015), 10367–10398. MR 3455870, DOI 10.1093/imrn/rnu254
- Thomas J. Haines, Correction to “On Satake parameters for representations with parahoric fixed vectors” [ MR3455870], Int. Math. Res. Not. IMRN 13 (2017), 4160–4170. MR 3671514, DOI 10.1093/imrn/rnx088
- Thomas J. Haines and Xuhua He, Vertexwise criteria for admissibility of alcoves, Amer. J. Math. 139 (2017), no. 3, 769–784. MR 3650232, DOI 10.1353/ajm.2017.0020
- Thomas J. Haines, Robert E. Kottwitz, and Amritanshu Prasad, Iwahori-Hecke algebras, J. Ramanujan Math. Soc. 25 (2010), no. 2, 113–145. MR 2642451
- T. Haines and B. C. Ngô, Nearby cycles for local models of some Shimura varieties, Compositio Math. 133 (2002), no. 2, 117–150. MR 1923579, DOI 10.1023/A:1019666710051
- T. Haines and M. Rapoport, On parahoric subgroups, Advances in Math. 219 (2008), 188-198; appendix to: G. Pappas, M. Rapoport, Twisted loop groups and their affine flag varieties, Advances in Math. 219 (2008), 118-198.
- Thomas J. Haines and Michael Rapoport, Shimura varieties with $\Gamma _1(p)$-level via Hecke algebra isomorphisms: the Drinfeld case, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), no. 5, 719–785 (2013) (English, with English and French summaries). MR 3053008, DOI 10.24033/asens.2177
- T. Haines and T. Richarz, The test function conjecture for parahoric local models, preprint 2018. arXiv:1801.07094.
- Jiuzu Hong, Mirković-Vilonen cycles and polytopes for a symmetric pair, Represent. Theory 13 (2009), 19–32. MR 2480386, DOI 10.1090/S1088-4165-09-00341-0
- Jens Carsten Jantzen, Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen, Bonn. Math. Schr. 67 (1973), v+124. MR 401935
- Shin-ichi Kato, Spherical functions and a $q$-analogue of Kostant’s weight multiplicity formula, Invent. Math. 66 (1982), no. 3, 461–468. MR 662602, DOI 10.1007/BF01389223
- David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153–215. MR 862716, DOI 10.1007/BF01389157
- M. Kisin and G. Pappas, Integral models of Shimura varieties with parahoric level structure, arXiv:1512.01149.
- Friedrich Knop, On the Kazhdan-Lusztig basis of a spherical Hecke algebra, Represent. Theory 9 (2005), 417–425. MR 2142817, DOI 10.1090/S1088-4165-05-00237-2
- Robert E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611–650. MR 757954, DOI 10.1215/S0012-7094-84-05129-9
- Robert E. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), no. 3, 255–339. MR 1485921, DOI 10.1023/A:1000102604688
- Robert E. Kottwitz and Diana Shelstad, Foundations of twisted endoscopy, Astérisque 255 (1999), vi+190 (English, with English and French summaries). MR 1687096
- Shrawan Kumar, George Lusztig, and Dipendra Prasad, Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups, Representation theory, Contemp. Math., vol. 478, Amer. Math. Soc., Providence, RI, 2009, pp. 99–101. MR 2513268, DOI 10.1090/conm/478/09321
- Erasmus Landvogt, A compactification of the Bruhat-Tits building, Lecture Notes in Mathematics, vol. 1619, Springer-Verlag, Berlin, 1996. MR 1441308, DOI 10.1007/BFb0094594
- Brandon Levin, Local models for Weil-restricted groups, Compos. Math. 152 (2016), no. 12, 2563–2601. MR 3594288, DOI 10.1112/S0010437X1600765X
- George Lusztig, Singularities, character formulas, and a $q$-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 208–229. MR 737932
- G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442, DOI 10.1090/crmm/018
- I. G. Macdonald, Spherical functions on a group of $p$-adic type, Publications of the Ramanujan Institute, No. 2, University of Madras, Centre for Advanced Study in Mathematics, Ramanujan Institute, Madras, 1971. MR 0435301
- J. Michel, Lectures on Coxeter groups, Lectures in Beijing, April-May 2014. Available at: https://webusers.imj-prg.fr/ jean.michel/.
- G. Pappas and M. Rapoport, Local models in the ramified case. I. The EL-case, J. Algebraic Geom. 12 (2003), no. 1, 107–145. MR 1948687, DOI 10.1090/S1056-3911-02-00334-X
- G. Pappas and M. Rapoport, Local models in the ramified case. II. Splitting models, Duke Math. J. 127 (2005), no. 2, 193–250. MR 2130412, DOI 10.1215/S0012-7094-04-12721-6
- G. Pappas and M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), no. 1, 118–198. With an appendix by T. Haines and Rapoport. MR 2435422, DOI 10.1016/j.aim.2008.04.006
- Georgios Pappas, Michael Rapoport, and Brian Smithling, Local models of Shimura varieties, I. Geometry and combinatorics, Handbook of moduli. Vol. III, Adv. Lect. Math. (ALM), vol. 26, Int. Press, Somerville, MA, 2013, pp. 135–217. MR 3135437
- Gopal Prasad and M. S. Raghunathan, Topological central extensions of semisimple groups over local fields, Ann. of Math. (2) 119 (1984), no. 1, 143–201. MR 736564, DOI 10.2307/2006967
- Michael Rapoport, A guide to the reduction modulo $p$ of Shimura varieties, Astérisque 298 (2005), 271–318 (English, with English and French summaries). Automorphic forms. I. MR 2141705
- M. Rapoport and Th. Zink, Period spaces for $p$-divisible groups, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996. MR 1393439, DOI 10.1515/9781400882601
- Timo Richarz, Schubert varieties in twisted affine flag varieties and local models, J. Algebra 375 (2013), 121–147. MR 2998951, DOI 10.1016/j.jalgebra.2012.11.013
- Timo Richarz, Affine Grassmannians and geometric Satake equivalences, Int. Math. Res. Not. IMRN 12 (2016), 3717–3767. MR 3544618, DOI 10.1093/imrn/rnv226
- T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4
- J. Tits, Reductive groups over local fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588
- Xinwen Zhu, The geometric Satake correspondence for ramified groups, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 2, 409–451 (English, with English and French summaries). MR 3346175, DOI 10.24033/asens.2248
- Xinwen Zhu, On the coherence conjecture of Pappas and Rapoport, Ann. of Math. (2) 180 (2014), no. 1, 1–85. MR 3194811, DOI 10.4007/annals.2014.180.1.1
Additional Information
- Thomas J. Haines
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742-4015
- MR Author ID: 659516
- Email: tjh@math.umd.edu
- Received by editor(s): June 29, 2016
- Received by editor(s) in revised form: January 23, 2018
- Published electronically: March 9, 2018
- Additional Notes: The author’s research was partially supported by NSF DMS-1406787
- © Copyright 2018 American Mathematical Society
- Journal: Represent. Theory 22 (2018), 1-26
- MSC (2010): Primary 20C08, 20G25, 22E50, 17B20; Secondary 11F70, 11G18
- DOI: https://doi.org/10.1090/ert/512
- MathSciNet review: 3772644