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4 - The stable Bernstein center and test functions for Shimura varieties

Published online by Cambridge University Press:  05 October 2014

Thomas J. Haines
Univeristy of Maryland
Fred Diamond
King's College London
Payman L. Kassaei
King's College London
Minhyong Kim
University of Oxford
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Publisher: Cambridge University Press
Print publication year: 2014

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[AC] J., Arthur, L., Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Math. Studies 120, Princeton University Press, 1989.
[BD] J.-N., Bernstein, rédigé par P. Deligne, Le “centre” de Bernstein, In: Représentations des groupes réductifs sur un corps local, Hermann (1984).
[Be92] J., Bernstein, Representations of p-adic groups, Notes taken by K. Rumel-hart of lectures by J. Bernstein at Harvard in the Fall of 1992.
[Bo79] A., Borel, Automorphic L-functions, In: Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math., vol. 33, part 2, Amer. Math. Soc., 1979, pp. 27–61.
[BT2] F., Bruhat, J., Tits, Groupes réductifs sur un corps local. II, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 5–184.Google Scholar
[BK] C. J., Bushnell, P. C., Kutzko, Smooth representations of reductive p-adic groups: structure theory via types, Proc. London Math. Soc. (3) 77 (1998), 582–634.Google Scholar
[Cas] W., Casselman, Characters and Jacquet modules, Math. Ann. 230 (1977), 101–105.Google Scholar
[Cl90] L., Clozel, The fundamental lemma for stable base change, Duke Math. J. 61, no. 1, (1990), 255–302.Google Scholar
[Fer] A., Ferrari, Théorème de l'indice et formule des traces, Manuscripta Math. 124, (2007), 363–390.Google Scholar
[Ga] D., Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math. 144, no. 2, (2001) 253–280.Google Scholar
[G] U., Görtz, Alcove walks and nearby cycles on affine flag manifolds, J. Alg. Comb. 26, no. 4 (2007), 415–430.Google Scholar
[GH] U., Görtz, T., Haines, The Jordan-Hoelder series for nearby cycles on some Shimura varieties and affine flag varieties, J. Reine Angew. Math. 609 (2007), 161–213.Google Scholar
[GR] B. H., Gross, M., Reeder, Arithmetic invariants of discrete Langlands parametersDuke Math. J. 154, no. 3, (2010), 431–508.Google Scholar
[H01] T., Haines, Test functions for Shimura varieties: the Drinfeld case, Duke Math. J. 106 no. 1, (2001), 19–40.Google Scholar
[H05] T., Haines, Introduction to Shimura varieties with bad reduction of parahoric type, Clay Math. Proc. 4, (2005), 583–642.Google Scholar
[H07] T., Haines, Intertwiners for unramified groups, expository note (2007). Available at∼tjh.
[H09] T., Haines, The base change fundamental lemma for central elements in parahoric Hecke algebras, Duke Math. J., 149, no. 3 (2009), 569–643.Google Scholar
[H11] T., Haines, Endoscopic transfer of the Bernstein center, IAS/Princeton Number theory seminar, April 6, 2011. Slides available at∼tjh.
[H12] T., Haines, Base change for Bernstein centers of depth zero principal series blocks, Ann. Scient. École Norm. Sup. 4et. 45, 2012, 681–718.Google Scholar
[HKP] T., Haines, R., Kottwitz, A., Prasad, Iwahori-Hecke algebras, J. Ramanujan Math. Soc. 25, no. 2 (2010), 113–145.Google Scholar
[HN02a] T., Haines, B.C., Ngô, Nearby cycles for local models of some Shimura varieties, Compo. Math. 133, (2002), 117–150.Google Scholar
[HN02b] T., Haines, B. C., Ngô, Alcoves associated to special fibers of local models, Amer. J. Math. 124 (2002), 11251152.Google Scholar
[HP] T., Haines, A., Pettet, Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra, J. Alg. 252 (2002), 127–149.Google Scholar
[HRa1] T., Haines, M., Rapoport, Shimura varieties with Γ1 (p)-level via Hecke algebra isomorphisms: the Drinfeld case, Ann. Scient. École Norm. Sup. 4e série, t. 45, (2012), 719–785.Google Scholar
[HRo] T., Haines, S., Rostami, The Satake isomorphism for special maximal parahoric Hecke algebras, Represent. Theory 14 (2010), 264–284.Google Scholar
[Hal] T., Hales, On the fundamental lemma for standard endoscopy: reduction to unit elements, Canad. J. Math., 47(5), (1995), 974–994.Google Scholar
[Kal] T., Kaletha, Epipelagic L-packets and rectifying characters. Preprint (2012). arXiv:1209.1720.
[Kaz] D., Kazhdan, Cuspidal geometry of p-adic groups, J. Analyse Math., 47, (1986), 1–36.Google Scholar
[KL] D., Kazhdan and G., Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184.Google Scholar
[KV] D., Kazhdan, Y., Varshavsky, On endoscopic transfer of Deligne-Lusztig functions, Duke Math. J. 161, no. 4, (2012), 675–732.Google Scholar
[Ko84a] R., Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51, no. 3, (1984), 611–650.Google Scholar
[Ko84b] R., Kottwitz, Shimura varieties and twisted orbital integrals, Math. Ann. 269 (1984), 287–300.Google Scholar
[Ko86] R., Kottwitz, Base change for unit elements of Hecke algebras, Comp. Math. 60, (1986), 237–250.Google Scholar
[Ko88] R., Kottwitz, Tamagawa numbers, Ann. of Math. 127 (1988), 629–646.Google Scholar
[Ko90] R., Kottwitz, Shimura varieties and λ-adic representations, in Automorphic forms, Shimura varieties and L-functions. Proceedings of the Ann Arbor conference, Academic Press, 1990.
[Ko92a] R., Kottwitz, Points of some Shimura varieties over finite fields, J. Amer. Math. Soc. 5, (1992), 373–444.Google Scholar
[Ko92b] R., Kottwitz, On the λ-adic representations associated to some simple Shimura varieties, Invent. Math. 108, (1992), 653–665.Google Scholar
[Ko97] R., Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109, (1997), 255-339.Google Scholar
[KR] R., Kottwitz, M., Rapoport, Minuscule alcoves for GLn and GSP2n, Manuscripta Math. 102, no. 4, (2000), 403–428.Google Scholar
[Land] E., Landvogt, A compactification of the Bruhat-Tits building, Lecture Notes in Mathematics 1619, Springer, 1996.
[L1] R. P., Langlands, Shimura varieties and the Selberg trace formula, Can. J. Math. 29, (1977), 1292–1299.Google Scholar
[L2] R. P., Langlands, On the zeta-functions of some simple Shimura varieties, Can. J. Math. 31, (1979), 1121–1216.Google Scholar
[Mil] J. S., Milne, The points on a Shimura variety modulo a prime of good reduction, In: The zeta functions of Picard modular surfaces, ed. R. P., Landlands and D., Ramakrishnan, CRM, 1992, 151–253.
[MV] I., Mirković, K., Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166, no. 1, (2007), 95–143.Google Scholar
[Mor] S., Morel, On the cohomology of certain non-compact Shimura varieties, Annals of Math. Studies 173, Princeton University Press, 2010.
[Ngo] B. C., Ngô, Le lemme fondamental pour les algèbres de Lie, Publ. Math. IHÉS 111, (2010), 1–169.Google Scholar
[PRS] G., Pappas, M., Rapoport, B., Smithling, Local models of Shimura varieties, I. Geometry and combinatorics, to appear in the Handbook of Moduli, 84 pp.
[PZ] G., Pappas, X., Zhu, Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math. 194, no. 1 (2013), 147–254.Google Scholar
[PrRa] Prasad, D., Ramakrishnan, D.Self-dual representations of division algebras and Weil groups: a contrast, Amer. J. Math. 134, no. 3, (2012), 749–767.Google Scholar
[Ra90] M., Rapoport, On the bad reduction of Shimura varieties, in Automorphic forms, Shimura varieties and L-functions. Proceedings of the Ann Arbor conference, Academic Press, 1990.
[Ra05] M., Rapoport: A guide to the reduction modulo p of Shimura varieties. Astérisque 298, (2005), 271–318.Google Scholar
[RZ] M., Rapoport, T., Zink, Period spaces for p-divisible groups, Annals of Math. Studies 141, Princeton University Press, 1996.
[Ren] D., Renard, Représentations des groupes réductifs p-adiques. Cours Spécialisés, 17. Société Mathématique de France, 2010.
[Ri] T., Richarz, On the Iwahori-Weyl group, available at
[Roc] A., Roche, The Bernstein decomposition and the Bernstein centre, In: Ottawa Lectures on admissible representations of reductive p-adic groups, Fields Inst. Monogr. 26, Amer. Math. Soc., 2009, pp. 3–52.
[Ro] S., Rostami, The Bernstein presentation for general connected reductive groups, arXiv:1312.7374.
[Sch1] P., Scholze, The Langlands-Kottwitz approach for the modular curve, IMRN 2011, no. 15, 3368–3425.
[Sch2] P., Scholze, The Langlands-Kottwitz approach for some simple Shimura varieties. Invent. Math. 192, no. 3, 627–661.
[Sch3] P., Scholze, The local Langlands correspondence for GLn over p-adic fields. Invent. Math. 192, no. 3, 663–715.
[Sch4] P., Scholze, The Langlands-Kottwitz method and deformation spaces of p-divisible groups, J. Amer. Math. Soc. 26 (2013), 227-259.Google Scholar
[SS] P., Scholze, S. W., Shin, On the cohomology of compact unitary group Shimura varieties at ramified split places, J. Amer. Math. Soc. 26 (2013), 261–294.Google Scholar
[Sh] S. W., Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. (2) 173, no. 3, (2011), 1645–1741.Google Scholar
[Sm1] B., Smithling, Topological flatness of orthogonal local models in the split, even case, I. Math. Ann. 35, no. 2, (2011), 381–416.Google Scholar
[Sm2] B., Smithling, Admissibility and permissibility for minuscule cocharacters in orthogonal groups, Manuscripta Math. 136, no. 3–4, (2011), 295–314.Google Scholar
[Sm3] B., Smithling, Topological flatness of local models for ramified unitary groups. I. The odd dimensional case, Adv. Math. 226, no. 4, (2011), 31603190.Google Scholar
[St] R., Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80 (1968), 1–108.Google Scholar
[Str] B., Stroh, Sur une conjecture de Kottwitz au bord, Ann. Sci. Ec. Norm. Sup. (4) 45, no. 1, (2012), 143–165.Google Scholar
[Var] S., Varma, On certain elements in the Bernstein center of GL2, Represent. Theory 17, (2013), 99–119.Google Scholar
[Vo] D., Vogan, The local Langlands conjecture, In: Representation theory of groups and algebras, Contemp. Math. 145, 1993, pp. 305–379.
[Wal97] J.-L., Waldspurger, Le lemme fondamental implique le transfert, Compositio Math. 105, (1997), 153–236.Google Scholar
[Wal04] J.-L., Waldspurger, Endoscopie et changement de caractéristique, J. Inst. Math. Jussieu 5, no. 3, (2006), 423–525.Google Scholar
[Wal08] J.-L., Waldspurger, L'endoscopie tordue n'est pas si tordue, Mem. Amer. Math. Soc. 194, no. 198, 2008.Google Scholar
[Zhu] X., Zhu, The geometric Satake correspondence for ramified groups, arXiv:1107.5762.
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