×

zbMATH — the first resource for mathematics

Induced representations and classifications for \(GSp(2,F)\) and \(Sp(2,F)\). (English) Zbl 0784.22008
Let \(F\) be a \(p\)-adic field whose characteristic is different from 2. The authors study the reducibility of the representations of the groups \(\text{GSp}(2,F)\) and \(\text{Sp}(2,F)\) induced by irreducible representations of their parabolic subgroups. This leads to the classification results for various classes of irreducible representations (modulo cuspidal representations). In particular, the classification of all irreducible unitary representations is given.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
22D30 Induced representations for locally compact groups
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Bernstein, J. and ZELEVINSKY, A.V. , Induced representations of reductive p-adic groups I , Ann. Sci. École Norm Sup. 10 ( 1977 ), 441-472. Numdam | MR 58 #28310 | Zbl 0412.22015 · Zbl 0412.22015 · numdam:ASENS_1977_4_10_4_441_0 · eudml:82002
[2] Borel, A. and Wallach, N. , Continuous cohomology, discrete subgroups, and representations of reductive groups , Princeton University Press, Princeton, ( 1980 ). MR 83c:22018 | Zbl 0443.22010 · Zbl 0443.22010
[3] Casselman, W. , Introduction to the theory of admissible representations of p-adic reductive groups , preprint.
[4] Gelbart, S.S. and Knapp, A. W. , L-indistinguishability and R groups for the special linear group , Advan. in Math., 43 ( 1982 ), 101-121. MR 83j:22009 | Zbl 0493.22005 · Zbl 0493.22005 · doi:10.1016/0001-8708(82)90030-5
[5] Gustafson, R. , The degenerate principal series for Sp(2n) , Mem. of the Amer. Math. Society, 248 ( 1981 ), 1-81. MR 83e:22021 | Zbl 0482.22013 · Zbl 0482.22013
[6] Howe, R. and Moore, C.C. , Asymptotic properties of unitary representations , J. Functional Analysis, 32, No. 1 ( 1979 ), 72-96. MR 80g:22017 | Zbl 0404.22015 · Zbl 0404.22015 · doi:10.1016/0022-1236(79)90078-8
[7] Jantzen, C. , Degenerate principal series for symplectic groups , to appear in Mem. of the Amer. Math. Society. Zbl 0814.22004 · Zbl 0814.22004
[8] Keys, D. , On the decomposition of reducible principal series representations of p-adic Chevalley groups , Pacific J. Math, 101 ( 1982 ), 351-388. Article | MR 84d:22032 | Zbl 0438.22010 · Zbl 0438.22010 · doi:10.2140/pjm.1982.101.351 · minidml.mathdoc.fr
[9] Milici, D. , On C*-algebras with bounded trace , Glasnik Mat., 8(28) ( 1973 ), 7-21. MR 48 #2781 | Zbl 0265.46072 · Zbl 0265.46072
[10] Moy, A. , Representations of GSp(4) over a p-adic field : parts 1 and 2 , Compositio Math., 66 ( 1988 ), 237-328. Numdam | MR 90d:22022 | Zbl 0662.22012 · Zbl 0662.22012 · numdam:CM_1988__66_3_237_0 · numdam:CM_1988__66_3_285_0 · eudml:89904
[11] Rodier, F. , Décomposition de la série principale des groupes réductifs p-adiques , Non-Commutative Harmonic Analysis, Lecture Notes in Math. 880, Springer-Verlag, Berlin ( 1981 ). MR 83i:22029 | Zbl 0465.22009 · Zbl 0465.22009
[12] Rodier, F. Sur les représentations non ramifiées des groupes réductifs p-adiques ; l’example de GSp(4) , Bull. Soc. Math. France, 116 ( 1988 ), 15-42. Numdam | MR 89i:22033 | Zbl 0662.22011 · Zbl 0662.22011 · numdam:BSMF_1988__116_1_15_0 · eudml:87546
[13] Shahidi, F. , A proof of Langlands conjecture on Plancherel measures ; complementary series for p-adic groups , Ann. of Math., 132 ( 1990 ), 273-330. MR 91m:11095 | Zbl 0780.22005 · Zbl 0780.22005 · doi:10.2307/1971524
[14] Shahidi, F. , Langlands’ conjecture on Plancherel measures of p-adic groups , preprint. · Zbl 0852.22017
[15] Shahidi, F. , L-functions and representation theory of p-adic , preprint. · Zbl 0780.22006
[16] Shahidi, F. , Letter .
[17] Tadić, M. , Classification of unitary representations in irreducible representations of general linear group (non-archimedean case) , Ann. Sci. École Norm. Sup, 19 ( 1986 ), 335-382. Numdam | MR 88b:22021 | Zbl 0614.22005 · Zbl 0614.22005 · numdam:ASENS_1986_4_19_3_335_0 · eudml:82179
[18] Tadić, M. Induced representations of GL(n, A) for p-adic division algebras A , J. reine angew. Math., 405 ( 1990 ), 48-77. MR 91i:22025 | Zbl 0684.22008 · Zbl 0684.22008 · doi:10.1515/crll.1990.405.48 · crelle:GDZPPN002207362 · eudml:153209
[19] Tadić, M. , Notes on representations of non-archimedean SL(n) , Pacific J. Math. (to appear). · Zbl 0724.22017
[20] Tadić, M. , Representations of p-adic symplectic groups , preprint. · Zbl 0797.22008
[21] Tadić, M. , On Jaquet modules of induced representations of p-adic symplectic groups , preprint. · Zbl 0760.22008
[22] Tadić, M. , A structure arising from induction and restriction of representations of classical p-adic groups , preprint. · Zbl 0874.22014 · doi:10.1006/jabr.1995.1284
[23] Tadić, M. , Representations of classical p-adic groups , preprint. · Zbl 0859.22011
[24] Waldspurger, J.-L. , Un exercice sur GSp(4, F) et les représentations de Weil , Bull. Soc. Math. France, 115 ( 1987 ), 35-69. Numdam | MR 89a:22033 | Zbl 0635.22016 · Zbl 0635.22016 · numdam:BSMF_1987__115__35_0 · eudml:87537
[25] Zelevinsky, A.V. , Induced representations of reductive p-adic groups II, On irreducible representations of GL(n) , Ann. Sci École Norm Sup., 13 ( 1980 ), 165-210. Numdam | MR 83g:22012 | Zbl 0441.22014 · Zbl 0441.22014 · numdam:ASENS_1980_4_13_2_165_0 · eudml:82048
[26] Zelevinsky, A.V. , Representations of Finite Classical Groups, A Hopf Algebra Approach , Lecture Notes in Math 869, Springer-Verlag, Berlin, ( 1981 ). MR 83k:20017 | Zbl 0465.20009 · Zbl 0465.20009 · doi:10.1007/BFb0090287
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.