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 Representation Theory
Representation Theory
ISSN 1088-4165

     

Jacquet modules of $ p$-adic general linear groups

Author(s): Chris Jantzen
Journal: Represent. Theory 11 (2007), 45-83.
MSC (2000): Primary 22E50
Posted: April 18, 2007
MathSciNet review: 2306606
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Abstract | References | Similar articles | Additional information

Abstract: In this paper, we study Jacquet modules for $ p$-adic general linear groups. More precisely, we have results--formulas and algorithms--aimed at addressing the following question: Given the Langlands data for an irreducible representation, can we determine its (semisimplified) Jacquet module? We use our results to answer this question in a number of cases, as well as to recover some familiar results as relatively easy consequences.

References:

[Aub]
A.-M. Aubert, Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d'un groupe réductif $ p$-adique, Trans. Amer. Math. Soc., 347(1995), 2179-2189, and Erratum à ``Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d'un groupe réductif $ p$-adique'', Trans. Amer. Math. Soc., 348(1996), 4687-4690. MR 1285969 (95i:22025)

[B-Z]
I. Bernstein and A. Zelevinsky, Induced representations of reductive $ p$-adic groups $ I$, Ann. Sci. École Norm. Sup., 10 (1977), 441-472. MR 0579172 (58:28310)

[B-W]
A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Princeton University Press, Princeton, 1980. MR 554917 (83c:22018)

[B-K]
C. Bushnell and P. Kutzko, The Admissible Dual of $ GL(N)$ Via Compact Open Subgroups, Princeton University Press, Princeton, 1993. MR 1204652 (94h:22007)

[Cas1]
W. Casselman, Introduction to the theory of admissible representations of $ p$-adic reductive groups, preprint.

[Cas2]
W. Casselman, The Steinberg character as a true character, Proc. Sympos. Pure Math., 26(1973), 413-417. MR 0338273 (49:3039)

[Jan1]
C. Jantzen, Degenerate principal series for symplectic groups, Mem. Amer. Math. Soc., 488(1993), 1-111. MR 1134591 (93g:22018)

[Jan2]
C. Jantzen, On supports of induced representations for symplectic and odd-orthogonal groups, Amer. J. Math., 119(1997), 1213-1262. MR 1481814 (99b:22028)

[Jan3]
C. Jantzen, Some remarks on degenerate principal series, Pacific J. Math., 186(1998), 67-87. MR 1665057 (99j:22018)

[Jan4]
C. Jantzen, Square-integrable representations for symplectic and odd-orthogonal groups, Canad. J. Math., 52(2000), 539-581. MR 1758232 (2001f:22056)

[Kat]
S.-I. Kato, Duality for representations of a Hecke algebra, Proc. Amer. Math. Soc., 119(1993), 941-946. MR 1215028 (94g:20060)

[K-Z]
H. Knight and A. Zelevinsky, Representations of quivers of type A and multisegment duality, Adv. Math., 117(1996), 273-293. MR 1371654 (97e:16029)

[Kon]
T. Konno, A note on the Langlands classification and irreducibility of induced representations of $ p$-adic groups, Kyushu J. Math., 57(2003), 383-409. MR 2050093 (2005b:22020)

[M-W]
C. M\oeglin and J.-L. Waldspurger, Sur l'involution de Zelevinski, J. Reine Angew. Math., 372(1986), 136-177. MR 863522 (88c:22019)

[S-S]
P. Schneider and U. Stuhler, Representation theory and sheaves on the Bruhat-Tits building, Publ. Math. IHES, 85(1997), 97-191. MR 1471867 (98m:22023)

[Sil]
A. Silberger, The Langlands quotient theorem for $ p$-adic groups, Math. Ann., 236(1978), 95-104. MR 0507262 (58:22413)

[Tad1]
M. Tadic, Induced representations of $ GL(n,A)$ for $ p$-adic division algebras $ A$, J. Reine Angew. Math., 405(1990), 48-77. MR 1040995 (91i:22025)

[Tad2]
M. Tadic, Structure arising from induction and Jacquet modules of representations of classical $ p$-adic groups, J. Algebra, 177(1995), 1-33. MR 1356358 (97b:22023)

[Tad3]
M. Tadic, On reducibility of parabolic induction, Israel J. Math., 107(1998), 29-91. MR 1658535 (2001d:22012)

[Zel]
A. Zelevinsky, Induced representations of reductive $ p$-adic groups $ II$, On irreducible representations of $ GL(n)$, Ann. Sci. École Norm. Sup., 13 (1980), 165-210. MR 584084 (83g:22012)

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Additional Information:

Chris Jantzen
Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
Email: jantzenc@ecu.edu

DOI: 10.1090/S1088-4165-07-00316-0
PII: S 1088-4165(07)00316-0
Received by editor(s): October 11, 2006
Posted: April 18, 2007
Additional Notes: This research was supported in part by NSA grant H98230-04-1-0029 and the East Carolina University College of Arts and Sciences
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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