×

The residual spectrum of \(Sp_ 4\). (English) Zbl 0877.11030

In this paper the author uses Langlands’ method to analyze the residual spectrum of the group \(Sp_4\) over a number field. According to Langlands’ general principles there is a decomposition corresponding to the classes of parabolic subgroups. In the case of the Siegel parabolic subgroup the author obtains a decomposition depending on cuspidal representations of \(GL_2\) with trivial central characters satisfying, essentially, \(L(1/2,\pi) \neq 0\). In the case of the other two maximal parabolic subgroups he obtains a decomposition parametrized by monomial representations of \(GL_2\). In the case of the Borel subgroup the decomposition is parametrized by Größencharaktere of order 2, but the irreducible representations are selected by a parity condition on the \(\varepsilon\)-factors and so do not correspond to the entire global \(L\)-packet.

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] Arthur, J. : On some problems suggested by the trace formula, in Lie group representations, II , Lecture Notes in Math. 1041 Springer-Verlag, 1984, pp. 1-49. · Zbl 0541.22011
[2] Borel, A. and Wallach, N. : Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Math. Studies 94, Princeton University Press, Princeton, 1980. · Zbl 0443.22010
[3] Flicker, Y. : On the symmetric square: Applications of a trace formula , Trans. Amer. Math. Soc. 330(1) (1992), 125-152. · Zbl 0761.11027 · doi:10.2307/2154157
[4] Gelbart, S. : Automorphic forms on Adele groups, Annals of Math. Studies , No 83 Princeton University Press, Princeton, 1975. · Zbl 0329.10018 · doi:10.1515/9781400881611
[5] Gelbart, S. and Jacquet, H. : A relation between automorphic representations of GL(2) and GL(3) , Ann. Sci. Ecole Norm. Sup. 11 (1978), 471-542. · Zbl 0406.10022 · doi:10.24033/asens.1355
[6] Gelbart, S. and Shahidi, F. : Analytic Properties of Automorphic L-functions, Perspectives in Mathematics , Vol 6, Academic Press, New York, 1988. · Zbl 0654.10028
[7] Goldberg, D. : Reducibility of induced representations for Sp(2n) and SO (n) , to appear in Amer. J. Math. (1994). · Zbl 0851.22021 · doi:10.2307/2374942
[8] Howe, R. : Automorphic forms of low rank, in Non-Commutative Harmonic Analysis , Lecture Notes in Math. 880, Springer-Verlag, 1980, pp. 211-248. · Zbl 0463.10015
[9] Ikeda, T. : On the location of poles of the triple L-functions , Comp. Math. 83 (1992), 187-237. · Zbl 0773.11035
[10] Jacquet, H. : On the residual spectrum of GL(n), in Lie Group Representations II , Lecture Notes in Math. 1041, Springer-Verlag, 1983, pp. 185-208. · Zbl 0539.22016
[11] Jacquet, H. and Langlands, R.P. : Automorphic Forms on GL(2) , Lecture Notes in Math. 114, Springer-Verlag, 1970. · Zbl 0236.12010 · doi:10.1007/BFb0058988
[12] Keys, C.D. and Shahidi, F. : Artin L-functions and normalization of intertwining operators , Ann. Scient. Ec. Norm. Sup. 21 (1988), 67-89. · Zbl 0654.10030 · doi:10.24033/asens.1551
[13] Kudla, S. and Rallis, S. : Poles of Eisenstein series and L-functions, in Festschrift in Honor of I.I. Piatetski-Shapiro, Part II , Vol. 3, Proc. Israel Mathematical Conf., 1990, pp. 81-110. · Zbl 0712.11029
[14] Kudla, S. and Rallis, S. : On the Weil-Siegel formula , J. Reine Angew. Math. 387 (1988), pp. 1-68. · Zbl 0644.10021
[15] Kudla, S. , Rallis, S. , and Soudry, D. : On the degree 5 L-function for Sp(2) , Inv. Math. 107 (1992), 483-541. · Zbl 0776.11028 · doi:10.1007/BF01231900
[16] Labesse, J.P. and Langlands, R.P. : L-indistinguishability for SL(2) , Can. J. Math. 31(4) (1979), 726-785. · Zbl 0421.12014 · doi:10.4153/CJM-1979-070-3
[17] Langlands, R.P. : Euler Products , Yale University Press, Yale, 1971. · Zbl 0231.20016
[18] Langlands, R.P. : On the Functional Equations Satisfied by Eisenstein Series , Lecture Notes in Math. 544, Springer-Verlag, 1976. · Zbl 0332.10018 · doi:10.1007/BFb0079929
[19] Moeglin, C. : Orbites unipotentes et spectre discret non ramifie, Le cas des groupes classiques déployś , Comp. Math. 77 (1991), 1-54. · Zbl 0809.11030
[20] Moeglin, C. and Waldspurger, J.L. : Décomposition spectrale et séries d’Eisenstein, paraphrase sur l’E-criture, Progress in Mathematics , Birkhäuser, 1994. · Zbl 0794.11022
[21] Piatetski-Shapiro, I. and Rallis, I. : Rankin triple L functions , Comp. Math. 64 (1987), 31-115. · Zbl 0637.10023
[22] Piatetski-Shapiro, I. and Rallis, I. : A new way to get Euler products , J. Reine Angew. Math. 392 (1988), 110-124. · Zbl 0651.10021 · doi:10.1515/crll.1988.392.110
[23] Schwermer, J. : On Euler Products and Residual Eisenstein Cohomology Classes for Siegel Modular Varieties , Preprint, 1993. · Zbl 0829.11031 · doi:10.1515/form.1995.7.1
[24] Shahidi, F. : On certain L-functions , Amer. J. Math. 103 (1981), 297-356. · Zbl 0467.12013 · doi:10.2307/2374219
[25] Shahidi, F. : A proof of Langlands’ conjecture on Plancherel measure; complementary series for p-adic groups , Ann. Math. 132 (1990), 273-330. · Zbl 0780.22005 · doi:10.2307/1971524
[26] Shahidi, F. : Whittaker models for real groups , Duke Math. J. 47(1) (1980), 99-125. · Zbl 0433.22007 · doi:10.1215/S0012-7094-80-04708-0
[27] Shahidi, F. : On the Ramanujan conjecture and finiteness of poles for certain L-functions , Ann. Math. 127 (1988), 547-584. · Zbl 0654.10029 · doi:10.2307/2007005
[28] Shahidi, F. : Langlands’ conjecture on Plancherel measures for p-adic groups, in Harmonic Analysis on Reductive Groups, Progress in Math. 101, Birkhauser, Boston, 1991, pp. 277-295. · Zbl 0852.22017
[29] Silberger, A. : Introduction to Harmonic Analysis on Reductive p-adic Groups , Math. Notes 23, Princeton University Press, Princeton, 1979. · Zbl 0458.22006 · doi:10.1515/9781400871131
[30] Soudry, D. : The CAP representations of GSp(4, A) , J. Reine Angew. Math. 383 (1988), 87-108. · Zbl 0625.22013 · doi:10.1515/crll.1988.383.87
[31] Watanabe, T. : Residual automorphic representations of Sp 4 , Nagoya Math. J. 127 (1992), 15-47. · Zbl 0788.11019 · doi:10.1017/S0027763000004086
[32] Winarsky, N. : Reducibility of principal series representations of p-adic Chevalley groups , Amer. J. Math. 100(5) (1978), 941-956. · Zbl 0475.43005 · doi:10.2307/2373955
[33] Arthur, J. : Eisenstein series and the trace formula, in Proc. Symp. in Pure Mathematics, Part I , Vol. 33, 1979, pp. 253-274. · Zbl 0431.22016
[34] Goldstein, L.J. : Analytic Number Theory , Prentice-Hall Englewood Cliffs, NJ, 1971. · Zbl 0226.12001
[35] Kim, H. and Shahidi, F. : Quadratic unipotent Arthur parameters and residual spectrum of Sp2n , Preprint. · Zbl 0866.11036 · doi:10.1353/ajm.1996.0009
[36] Jantzen, C. : Degenerate principal series for symplectic and odd-orthogonal groups , Preprint, 1994. · Zbl 0866.22016
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.