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Representation Theory

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ISSN 1088-4165

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The Satake isomorphism for special maximal parahoric Hecke algebras
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by Thomas J. Haines and Sean Rostami PDF
Represent. Theory 14 (2010), 264-284 Request permission

Abstract:

Let $G$ denote a connected reductive group over a nonarchimedean local field $F$. Let $K$ denote a special maximal parahoric subgroup of $G(F)$. We establish a Satake isomorphism for the Hecke algebra $\mathcal {H}_K$ of $K$-bi-invariant compactly supported functions on $G(F)$. The key ingredient is a Cartan decomposition describing the double coset space $K\backslash G(F)/K$. As an application we define a transfer homomorphism $t: \mathcal {H}_{K^*}(G^*) \rightarrow \mathcal {H}_K(G)$ where $G^*$ is the quasi-split inner form of $G$. We also describe how our results relate to the treatment of Cartier [Car], where $K$ is replaced by a special maximal compact open subgroup $\widetilde {K} \subset G(F)$ and where a Satake isomorphism is established for the Hecke algebra $\mathcal {H}_{\widetilde {K}}$.
References
  • A. Borel, Automorphic $L$-functions, 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. 27–61. MR 546608
  • Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314
  • 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
  • F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 5-184.
  • F. Bruhat and J. Tits, Groupes algébriques sur un corps local. Chapitre III. Compléments et applications à la cohomologie galoisienne, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 3, 671–698 (French). MR 927605
  • P. Cartier, Representations of $p$-adic groups: a survey, 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. 111–155. MR 546593
  • Thomas J. Haines, Introduction to Shimura varieties with bad reduction of parahoric type, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 583–642. MR 2192017
  • Thomas J. Haines, The base change fundamental lemma for central elements in parahoric Hecke algebras, Duke Math. J. 149 (2009), no. 3, 569–643. MR 2553880, DOI 10.1215/00127094-2009-045
  • T. Haines, R. Kottwitz, A. Prasad, Iwahori-Hecke algebras, math.RT/0309168. Preprint.
  • 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
  • Martin Kneser, Galois-Kohomologie halbeinfacher algebraischer Gruppen über ${\mathfrak {p}}$-adischen Körpern. II, Math. Z. 89 (1965), 250–272 (German). MR 188219, DOI 10.1007/BF02116869
  • Robert E. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), no. 3, 255–339. MR 1485921, DOI 10.1023/A:1000102604688
  • N. Krämer, Local models for ramified unitary groups, Abh. Math. Sem. Univ. Hamburg 73 (2003), 67–80. MR 2028507, DOI 10.1007/BF02941269
  • 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
  • G. Pappas and M. Rapoport, Local models in the ramified case. III. Unitary groups, J. Inst. Math. Jussieu 8 (2009), no. 3, 507–564. MR 2516305, DOI 10.1017/S1474748009000139
  • 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
  • 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
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Additional Information
  • Thomas J. Haines
  • Affiliation: University of Maryland, Department of Mathematics, College Park, Maryland 20742-4015
  • MR Author ID: 659516
  • Email: tjh@math.umd.edu
  • Sean Rostami
  • Affiliation: University of Maryland, Department of Mathematics, College Park, Maryland 20742-4015
  • Email: srostami@math.umd.edu
  • Received by editor(s): October 17, 2009
  • Received by editor(s) in revised form: November 29, 2009
  • Published electronically: March 8, 2010
  • Additional Notes: The first author was partially supported by NSF Focused Research Grant DMS-0554254 and NSF Grant DMS-0901723, and by a University of Maryland GRB Semester Award.
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 14 (2010), 264-284
  • MSC (2010): Primary 11E95, 20G25; Secondary 22E20
  • DOI: https://doi.org/10.1090/S1088-4165-10-00370-5
  • MathSciNet review: 2602034