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On reducibility of parabolic induction. (English) Zbl 0914.22019
The subject of the paper is the investigation of reducibility of parabolically induced representations of \(\text{Sp}(2n,F)\) and \(\text{SO}(2n+1,F)\), where \(F\) is a non-archimedean local field, by applying parabolic restriction to these representations. There is a variety of general results in which reducibility of induced representations is related to reducibility of representations induced from cuspidal representations. From given concrete applications I mention the determination of the composition series for representations induced from a one-dimensional representation of certain maximal parabolic subgroups. The method of this paper was used later by C. Jantzen to complete the determination of the composition series of representations induced from a one-dimensional representation of an arbitrary parabolic subgroup.

22E50 Representations of Lie and linear algebraic groups over local fields
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[1] [Ad] J. D. Adler,Self-contragredient supercuspidal representations of GL n , Proceedings of the American mathematical Society125 (1997), 2471–2479. · Zbl 0886.22011
[2] [Au] A.-M. Aubert,Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif p-adique, Transactions of the American Mathematical Society347 (1995), 2179–2189 (andErratum, Transactions of the American Mathematical Society348 (1996), 4687–4690). · Zbl 0827.22005
[3] [B] J. Bernstein, Draft of:Representations of p-adic groups (Lectures at Harvard University, 1992, written by Karl E. Rumelhart).
[4] [BZ] I. N. Bernstein and A. V. Zelevinsky,Induced representations of reductive p-adic groups I, Annales Scientifiques de l’École Normale Supérieure10 (1977), 441–472. · Zbl 0412.22015
[5] [C1] W. Casselman,The Steinberg character as a true character, Symposia on Pure Mathematics 26, American Mathematical Society, Providence, Rhode Island, 1973, pp. 413–417. · Zbl 0289.22017
[6] [C2] W. Casselman,Introduction to the theory of admissible representations of p-adic reductive groups, preprint.
[7] [GbKn] S. S. Gelbart and A. W. Knapp,L-indistinguishability and R groups for the special linear group, Advances in Mathematics43 (1982), 101–121. · Zbl 0493.22005
[8] [GfKa] I. M. Gelfand and D. A. Kazhdan,Representations of GL(n, k), Lie Groups and their Representations, Halstead Press, Budapest, 1974, pp. 95–118.
[9] [Go] D. Goldberg,Reducibility of induced representations for SP(2n) and SO(n), American Journal of Mathematics116 (1994), 1101–1151. · Zbl 0851.22021
[10] [Gu] R. Gustafson,The degenerate principal series for Sp(2n), Memoirs of the American Mathematical Society248 (1981), 1–81. · Zbl 0482.22013
[11] [J1] C. Jantzen,Degenerate principal series for symplectic groups, Memoirs of the American Mathematical Society488 (1993), 1–110.
[12] [J2] C. Jantzen,Degenerate principal series for orthogonal groups, Journal für die Reine und Angewandte Mathematik441 (1993), 61–98. · Zbl 0776.22004
[13] [J3] C. Jantzen,Degenerate principal series for symplectic and odd-orthogonal groups Memoirs of the American Mathematical Society590 (1996), 1–100.
[14] [J4] C. Jantzen,Reducibility of certain representations for symplectic and oddorthogonal groups, Compositio Mathematica104 (1996), 55–63. · Zbl 0866.22017
[15] [KuRa] S. S. Kudla and S. Rallis,Ramified degenerate principal series representations for Sp(n), Israel Journal of Mathematics78 (1992), 209–256. · Zbl 0787.22019
[16] [Mg] C. Moeglin, Letter, February 1997.
[17] [MgVW] C. Moeglin, M.-F. Vigneéras and J.-L. Waldspurger,Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics1291, Springer-Verlag, Berlin, 1987.
[18] [Mi] G. Muić,Some results on square integrable representations; irreducibility of standard representations preprint. · Zbl 0909.22029
[19] [MrRp] F. Murnaghan and J. Repka,Reducibility of induced representations of split classical p-adic groups, Compositio Mathematica, to appear. · Zbl 0917.22009
[20] [Rd1] M. Reeder,Whittaker models and unipotent representations of p-adic groups, Mathematische Annalen308 (1997), 587–592. · Zbl 0874.22015
[21] [Rd2] M. Reeder,Hecke algebras and harmonic analysis on p-adic groups, American Journal of Mathematics119 (1997), 225–248. · Zbl 1012.22030
[22] [Rd3] M. Reeder, Letter, November 1996.
[23] [Ro1] F. Rodier,Décomposition de la série principale des groupes réductifs p-adiques, inHarmonic Analysis, Lecture Notes in Mathematics880, Springer-Verlag, Berlin, 1981.
[24] [Ro2] F. Rodier,Représentations de GL(n, k)où k est un corps p-adique, Séminaire Bourbaki no. 587 (1982), Astérisque92–93 (1982), 201–218.
[25] [Ro3] F. Rodier,Sur les représentations non ramifiées des groupes réductifs p-adiques; l’example de GSp(4), Bulletin de la Société Mathématique de France116 (1988), 15–42.
[26] [SnSt] P. Schneider and U. Stuhler,Representation theory and sheaves on the Bruhat-Tits building, Publications Mathématiques de l’Institut des Hautes Études Scientifiques85 (1997), 97–191. · Zbl 0892.22012
[27] [SaT] P. J. Sally and M. Tadić,Induced representations and classifications for GSp(2,F)and Sp(2,F), Mémoires de la Société Mathématique de France52 (1993), 75–133.
[28] [Sh1] F. Shahidi,A proof of Langlands conjecture on Plancherel measures; complementary series for p-adic groups, Annals of Mathematics132 (1990), 273–330. · Zbl 0780.22005
[29] [Sh2] F. Shahidi,Twisted endoscopy and reducibility of induced representations for p-adic groups, Duke Mathematical Journal66 (1992), 1–41. · Zbl 0785.22022
[30] [Si] A. Silberger,Discrete series and classifications for p-adic groups I, American Journal of Mathematics103 (1981), 1241–1321. · Zbl 0484.22026
[31] [T1] M. Tadić,Classification of unitary representations in irreducible representations of general linear group (non-archimedean case), Annales Scientifiques de l’École Normale Supérieure19 (1986), 335–382. · Zbl 0614.22005
[32] [T2] M. Tadić,Induced representations of GL(n, A)for p-adic division algebras A, Journal für die Reine und Angewandte Mathematik405 (1990), 48–77. · Zbl 0684.22008
[33] [T3] M. Tadić,Notes on representations of non-archimedean SL(n), Pacific Journal of Mathematics152 (1992), 375–396. · Zbl 0724.22017
[34] [T4] M. Tadić,On Jacquet modules of induced representations of p-adic symplectic groups, inHarmonic Analysis on Reductive Groups, Proceedings, Bowdoin College 1989, Progress in Mathematics 101, Birkhäuser, Boston, 1991, pp. 305–314.
[35] [T5] M. Tadić,Representations of p-adic symplectic groups, Compositio Mathematica90 (1994), 123–181. · Zbl 0797.22008
[36] [T6] M. Tadić,Structure arising from induction and Jacquet modules of representations of classical p-adic groups, Journal of Algebra177 (1995), 1–33. · Zbl 0874.22014
[37] [T7] M. Tadić,On regular square integrable representations of p-adic groups, American Journal of Mathematics120 (1998), 159–210. · Zbl 0903.22008
[38] [W] J.-L. Waldspurger,Un exercice sur GSp(4,F)et les représentations de Weil, Bulletin de la Société Mathématique de France115 (1987), 35–69.
[39] [Z] A. V. Zelevinsky,Induced representations of reductive p-adic groups II, On irreducible representations of GL(n), Annles Scientifiques de l’École Normale Supérieure13 (1980), 165–210. · Zbl 0441.22014
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