Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165

 

 

The Satake isomorphism for special maximal parahoric Hecke algebras


Authors: Thomas J. Haines and Sean Rostami
Journal: Represent. Theory 14 (2010), 264-284
MSC (2010): Primary 11E95, 20G25; Secondary 22E20
Published electronically: March 8, 2010
MathSciNet review: 2602034
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [Bo] 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
  • [Bou] 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
  • [BT1] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 0327923
  • [BT2] F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 5-184.
  • [BT3] 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
  • [Car] 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
  • [H05] 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
  • [H09] 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, 10.1215/00127094-2009-045
  • [HKP] T. Haines, R. Kottwitz, A. Prasad, Iwahori-Hecke algebras, math.RT/0309168. Preprint.
  • [HR] 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, 10.1016/j.aim.2008.04.006
  • [Kn] Martin Kneser, Galois-Kohomologie halbeinfacher algebraischer Gruppen über 𝔭-adischen Körpern. II, Math. Z. 89 (1965), 250–272 (German). MR 0188219
  • [Ko97] Robert E. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), no. 3, 255–339. MR 1485921, 10.1023/A:1000102604688
  • [Kr] N. Krämer, Local models for ramified unitary groups, Abh. Math. Sem. Univ. Hamburg 73 (2003), 67–80. MR 2028507, 10.1007/BF02941269
  • [Land] Erasmus Landvogt, A compactification of the Bruhat-Tits building, Lecture Notes in Mathematics, vol. 1619, Springer-Verlag, Berlin, 1996. MR 1441308
  • [PR] 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, 10.1017/S1474748009000139
  • [Rap] 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
  • [Tits] 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

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 11E95, 20G25, 22E20

Retrieve articles in all journals with MSC (2010): 11E95, 20G25, 22E20


Additional Information

Thomas J. Haines
Affiliation: University of Maryland, Department of Mathematics, College Park, Maryland 20742-4015
Email: tjh@math.umd.edu

Sean Rostami
Affiliation: University of Maryland, Department of Mathematics, College Park, Maryland 20742-4015
Email: srostami@math.umd.edu

DOI: https://doi.org/10.1090/S1088-4165-10-00370-5
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.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.