Abstract
Jacquet modules of a reducible parabolically induced representation of a reductivep-adic group reduce in a way consistent with the transitivity of Jacquet modules. This fact can be used for proving irreducibility of parabolically induced representations. Classical groups are particularly convenient for application of this method, since we have very good information about part of the representation theory of their Levi subgroups (general linear groups are factors of Levi subgroups, and therefore we can apply the Bernstein-Zelevinsky theory). In the paper, we apply this type of approach to the problem of determining reducibility of parabolically induced representations ofp-adic Sp(n) and SO(2n+1). We present also a method for getting Langlands parameters of irreducible subquotients. In general, we describe reducibility of certain generalized principal series (and some other interesting parabolically induced representations) in terms of the reducibility in the cuspidal case. When the cuspidal reducibility is known, we get explicit answers (for example, for representations supported in the minimal parabolic subgroups, the cuspidal reducibility is well-known rank one reducibility).
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References
[Ad] J. D. Adler,Self-contragredient supercuspidal representations of GL n , Proceedings of the American mathematical Society125 (1997), 2471–2479.
[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).
[B] J. Bernstein, Draft of:Representations of p-adic groups (Lectures at Harvard University, 1992, written by Karl E. Rumelhart).
[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.
[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.
[C2] W. Casselman,Introduction to the theory of admissible representations of p-adic reductive groups, preprint.
[GbKn] S. S. Gelbart and A. W. Knapp,L-indistinguishability and R groups for the special linear group, Advances in Mathematics43 (1982), 101–121.
[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.
[Go] D. Goldberg,Reducibility of induced representations for SP(2n) and SO(n), American Journal of Mathematics116 (1994), 1101–1151.
[Gu] R. Gustafson,The degenerate principal series for Sp(2n), Memoirs of the American Mathematical Society248 (1981), 1–81.
[J1] C. Jantzen,Degenerate principal series for symplectic groups, Memoirs of the American Mathematical Society488 (1993), 1–110.
[J2] C. Jantzen,Degenerate principal series for orthogonal groups, Journal für die Reine und Angewandte Mathematik441 (1993), 61–98.
[J3] C. Jantzen,Degenerate principal series for symplectic and odd-orthogonal groups Memoirs of the American Mathematical Society590 (1996), 1–100.
[J4] C. Jantzen,Reducibility of certain representations for symplectic and oddorthogonal groups, Compositio Mathematica104 (1996), 55–63.
[KuRa] S. S. Kudla and S. Rallis,Ramified degenerate principal series representations for Sp(n), Israel Journal of Mathematics78 (1992), 209–256.
[Mg] C. Mœglin, Letter, February 1997.
[MgVW] C. Mœglin, M.-F. Vigneéras and J.-L. Waldspurger,Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics1291, Springer-Verlag, Berlin, 1987.
[Mi] G. Muić,Some results on square integrable representations; irreducibility of standard representations preprint.
[MrRp] F. Murnaghan and J. Repka,Reducibility of induced representations of split classical p-adic groups, Compositio Mathematica, to appear.
[Rd1] M. Reeder,Whittaker models and unipotent representations of p-adic groups, Mathematische Annalen308 (1997), 587–592.
[Rd2] M. Reeder,Hecke algebras and harmonic analysis on p-adic groups, American Journal of Mathematics119 (1997), 225–248.
[Rd3] M. Reeder, Letter, November 1996.
[Ro1] F. Rodier,Décomposition de la série principale des groupes réductifs p-adiques, inHarmonic Analysis, Lecture Notes in Mathematics880, Springer-Verlag, Berlin, 1981.
[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.
[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.
[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.
[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.
[Sh1] F. Shahidi,A proof of Langlands conjecture on Plancherel measures; complementary series for p-adic groups, Annals of Mathematics132 (1990), 273–330.
[Sh2] F. Shahidi,Twisted endoscopy and reducibility of induced representations for p-adic groups, Duke Mathematical Journal66 (1992), 1–41.
[Si] A. Silberger,Discrete series and classifications for p-adic groups I, American Journal of Mathematics103 (1981), 1241–1321.
[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.
[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.
[T3] M. Tadić,Notes on representations of non-archimedean SL(n), Pacific Journal of Mathematics152 (1992), 375–396.
[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.
[T5] M. Tadić,Representations of p-adic symplectic groups, Compositio Mathematica90 (1994), 123–181.
[T6] M. Tadić,Structure arising from induction and Jacquet modules of representations of classical p-adic groups, Journal of Algebra177 (1995), 1–33.
[T7] M. Tadić,On regular square integrable representations of p-adic groups, American Journal of Mathematics120 (1998), 159–210.
[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.
[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.
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Tadić, M. On reducibility of parabolic induction. Isr. J. Math. 107, 29–91 (1998). https://doi.org/10.1007/BF02764004
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DOI: https://doi.org/10.1007/BF02764004