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 Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(e) ISSN 0894-0347(p)

     

Construction of tame supercuspidal representations

Author(s): Jiu-Kang Yu
Journal: J. Amer. Math. Soc. 14 (2001), 579-622.
MSC (2000): Primary 22E50, 11F70; Secondary 20G25
Posted: March 23, 2001
MathSciNet review: 1824988
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Abstract | References | Similar articles | Additional information

Abstract:

We give a quite general construction of irreducible supercuspidal representations and supercuspidal types (in the sense of Bushnell and Kutzko) of $p$-adic groups. In the tame case, the construction should include all known constructions, and it is expected that this gives all supercuspidal representations. We also give a conjectural Hecke algebra isomorphism, which can be used to analyze arbitrary irreducible admissible representations, following the ideas of Howe and Moy.

References:

[Ad]
J.D. Adler: Refined anisotropic $K$-types and supercuspidal representations, Pacific J. Math. 185, no. 1, 1-32 (1998) MR 2000f:22019

[AR]
J.D. Adler and A. Roche: An intertwining result for $p$-adic groups, Canad. J. Math. 52, no. 3, 449-467 (2000) CMP 2000:12

[BK1]
C.J. Bushnell and P.C. Kutzko: The admissible dual of $\operatorname{GL}(N)$ via compact open subgroups, Annals Math. Studies, Princeton Univ. Press (1993) MR 94h:22007

[BK2]
C.J. Bushnell and P.C. Kutzko: Smooth representations of reductive $p$-adic groups: structure theory via types, Proc. London Math. Soc. 77, no. 3, 582-634 (1998) MR 2000c:22014

[BT1]
F. Bruhat et J. Tits: Groupes réductifs sur un corps local, Chapitre I, Publ. Math. I.H.E.S. 41, 5-251 (1972) MR 48:6265

[BT2]
F. Bruhat et J. Tits: Groupes réductifs sur un corps local, Chapitre II, Publ. Math. I.H.E.S. 60, 197-376 (1984) MR 86c:20042

[Co]
L. Corwin, A construction of the supercuspidal representations of $\operatorname{GL}_n(F)$, $F$$p$-adic, Trans. Amer. Math. Soc. 337, 1-58 (1993) MR 93g:22017

[De]
S. DeBacker, On supercuspidal characters of $\operatorname{GL}_\ell$, $\ell$ a prime, Ph. D. thesis, The University of Chicago (1997)

[Ge]
P. Gérardin: Weil representations associated to finite fields, J. Algebra 46, 54-101 (1977) MR 57:470

[Ho]
R. Howe: Tamely ramified supercuspidal representations for $\operatorname{GL}_n$, Pacific. J. Math. 73, 437-460 (1977) MR 58:11241

[HM]
R. Howe and A. Moy: Hecke algebra isomorphisms for $\operatorname{GL}_n$ over a $p$-adic field, J. Algebra 131, 388-424 (1990) MR 91f:22031

[Hu]
J.E. Humphreys: Linear algebraic groups, Springer-Verlag (1975) MR 53:633

[Ki]
J. Kim, Hecke algebras of classical groups over $p$-adic fields and supercuspidal representations, Amer. J. Math. 121, 967-1029 (1999) CMP 2000:01

[KZ]
H. Koch and E.-W. Zink: Zur Korrespondenz von Darstellungen der Galoisgruppen und der zentralen Divisionsalgebren über lokalen Körpern (Der Zahme Fall), Math. Nachr. 98, 83-119 (1980) MR 83h:12025

[L]
George Lusztig: Classification of unipotent representations of simple $p$-adic groups, Internat. Math. Res. Notices 11, 517-589 (1995) MR 98b:22034

[Mo1]
L.E. Morris, Level zero $G$-types, Comp. Math. 118, 135-157 (1999) MR 2000g:22029

[Mo2]
L.E. Morris, Some tamely ramified supercuspidal representations of symplectic groups, Proc. London Math. Soc. 63, 519-551 (1990) MR 92i:22017

[Mo3]
L.E. Morris, Tamely ramified supercuspidal representations of classical groups I, II, Ann. Sci. Éc. Norm. Sup. 24, 705-738 (1991), 25, 233-274 (1992) MR 93c:22032; MR 93h:22032

[Mo4]
L.E. Morris, Tamely ramified intertwining algebras, Invent. Math. 114, 1-54 (1993) MR 94g:22035

[My]
A. Moy: Local constants and the tame Langlands correspondence, Amer. J. Math. 108, 863-930 (1986) MR 88b:11081

[MP1]
A. Moy and G. Prasad: Unrefined minimal $K$-types for $p$-adic groups, Inv. Math. 116, 393-408 (1994) MR 95f:22023

[MP2]
A. Moy and G. Prasad: Jaquet functors and unrefined minimal $K$-types, Comment. Math. Helvetici 71, 96-121 (1996) MR 97c:22021

[P]
G. Prasad, Galois-fixed points in the Bruhat-Tits building of a reductive group, Bulletin Soc. Math. France, to appear.

[Se]
J.-P. Serre: Linear representations of finite groups, Springer-Verlag (1977) MR 56:8675

[Sp]
T.A. Springer: Reductive groups, Automorphic forms, representations, and $L$-functions (A. Borel and W. Casselman, Eds.), v.1., 3-28 (1977) MR 80h:20062

[St]
R. Steinberg: Torsion in reductive groups, Adv. in Math. 15, 63-92 (1975) MR 50:7369

[Ti]
J. Tits: Reductive groups over local fields, Automorphic forms, representations, and $L$-functions (A. Borel and W. Casselman, Eds.), v.1., 29-69 (1977) MR 80h:20064

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Additional Information:

Jiu-Kang Yu
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: yu@math.princeton.edu, yu@math.umd.edu

DOI: 10.1090/S0894-0347-01-00363-0
PII: S 0894-0347(01)00363-0
Keywords: Supercuspidal representation, Hecke algebra
Received by editor(s): August 30, 1999
Received by editor(s) in revised form: November 13, 2000
Posted: March 23, 2001
Additional Notes: The author was supported in part by grant DMS 9801633 from the National Science Foundation.
Copyright of article: Copyright 2001, American Mathematical Society




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