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A Satake isomorphism in characteristic p

Part of: Representation theory of groups Lie groups

Published online by Cambridge University Press:  23 August 2010

Florian Herzig
Florian Herzig*
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston IL 60208-2730, USA (email: herzig@math.northwestern.edu)
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Abstract

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Suppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported,K-biequivariant functions f:G(F)→End   V. These Hecke algebras were first considered by Barthel and Livné for GL 2. They play a role in the recent mod p andp-adic Langlands correspondences for GL 2 (ℚp) , in generalisations of Serre’s conjecture on the modularity of mod p Galois representations, and in the classification of irreducible mod p representations of unramified p-adic reductive groups.

Keywords

Hecke algebrasSatake isomorphismcharacteristic p

MSC classification

Secondary: 20C08: Hecke algebras and their representations 22E50: Representations of Lie and linear algebraic groups over local fields

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 1 , January 2011 , pp. 263 - 283
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Barthel, L. and Livné, R., Irreducible modular representations of GL2 of a local field, Duke Math. J. 75 (1994), 261–292; MR 1290194(95g:22030).Google Scholar
[2]Barthel, L. and Livné, R., Modular representations of GL2 of a local field: the ordinary, unramified case, J. Number Theory 55 (1995), 1–27; MR 1361556(96m:22036).Google Scholar
[3]Borel, A., Linear algebraic groups, Graduate Texts in Mathematics, vol. 126, second edition (Springer, New York, 1991); MR 1102012(92d:20001).Google Scholar
[4]Bourbaki, N., Lie groups and Lie algebras, chapters 4–6, Elements of Mathematics (Springer, Berlin, 2002); MR 1890629(2003a:17001).Google Scholar
[5]Breuil, C., Sur quelques représentations modulaires et p-adiques de GL2(Qp). I, Compositio Math. 138 (2003), 165–188; MR 2018825(2004k:11062).Google Scholar
[6]Bruhat, F. and Tits, J., Groupes réductifs sur un corps local, Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–251; MR 0327923(48#6265).Google Scholar
[7]Bruhat, F. and Tits, J., Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Publ. Math. Inst. Hautes Études Sci. 60 (1984), 197–376; MR 756316(86c:20042).Google Scholar
[8]Cabanes, M., Irreducible modules and Levi supplements, J. Algebra 90 (1984), 84–97; MR 757083(86a:20047).Google Scholar
[9]Cartier, P., Representations of p-adic groups: a survey, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proceedings of Symposia in Pure Mathematics, vol. XXXIII (American Mathematical Society, Providence, RI, 1979), 111–155; MR 546593(81e:22029).Google Scholar
[10]Colliot-Thélène, J.-L., Résolutions flasques des groupes linéaires connexes, J. Reine Angew. Math. 618 (2008), 77–133; MR 2404747(2009a:11091).Google Scholar
[11]Gross, B. H., On the Satake isomorphism, in Galois representations in arithmetic algebraic geometry (Durham, 1996), London Mathematical Society Lecture Note Series, vol. 254 (Cambridge University Press, Cambridge, 1998), 223–237; MR 1696481(2000e:22008).Google Scholar
[12]Gross, B. H., Algebraic modular forms, Israel J. Math. 113 (1999), 61–93; MR 1729443(2001b:11037).Google Scholar
[13]Grothendieck, A.et al., SGA 3: Schémas en groupes I, II, III, Lecture Notes in Mathematics, vol. 151, 152, 153 (Springer, Heidelberg, 1970).Google Scholar
[14]Haines, T., Kottwitz, R. and Prasad, A., Iwahori–Hecke algebras, J. Ramanujan Math. Soc. 25 (2010), 113–145.Google Scholar
[15]Herzig, F., The classification of irreducible admissible mod p representations of a p-adic GLn, Preprint (2010), http://math.northwestern.edu/∼herzig/parab-ind.pdf.Google Scholar
[16]Herzig, F., The weight in a Serre-type conjecture for tame n-dimensional Galois representations, Duke Math. J. 149 (2009), 37–116; MR 2541127.Google Scholar
[17]Humphreys, J. E., Modular representations of finite groups of Lie type, London Mathematical Society Lecture Note Series, vol. 326 (Cambridge University Press, Cambridge, 2006); MR 2199819(2007f:20023).Google Scholar
[18]Jantzen, J. C., Representations of algebraic groups, Mathematical Surveys and Monographs, vol. 107, second edition (American Mathematical Society, Providence, RI, 2003); MR 2015057(2004h:20061).Google Scholar
[19]Kato, S., Spherical functions and a q-analogue of Kostant’s weight multiplicity formula, Invent. Math. 66 (1982), 461–468; MR 662602(84b:22030).Google Scholar
[20]Lazarus, X., Module universel en caractéristique l>0 associé à un caractère de l’algèbre de Hecke de GL(n) sur un corps p-adique, avec l≠p, J. Algebra 213 (1999), 662–686; MR 1673473(2000f:20004).Google Scholar
[21]Schein, M., Weights in generalizations of Serre’s conjecture and the mod p local Langlands correspondence, in Symmetries in algebra and number theory (Göttingen, 2008) (Georg-August-Universitätsverlag, Göttingen, 2009), 71–93.Google Scholar
[22]Schneider, P. and Teitelbaum, J., Banach–Hecke algebras and p-adic Galois representations, Doc. Math. Extra Vol. (2006), 631–684 (electronic); MR 2290601(2008b:11126).Google Scholar
[23]Smith, S. D., Irreducible modules and parabolic subgroups, J. Algebra 75 (1982), 286–289; MR 650422(83g:20043).Google Scholar
[24]Springer, T. A., Linear algebraic groups, second edition, Progress in Mathematics, vol. 9 (Birkhäuser, Boston, MA, 1998); MR 1642713(99h:20075).Google Scholar
[25]Tits, J., Reductive groups over local fields, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proceedings of Symposia in Pure Mathematics, vol. XXXIII (American Mathematical Society, Providence, RI, 1979), 29–69; MR 546588(80h:20064).Google Scholar