The residual spectrum of $U(2,2)$
HTML articles powered by AMS MathViewer
- by Takyua Kon-No
- Trans. Amer. Math. Soc. 350 (1998), 1285-1358
- DOI: https://doi.org/10.1090/S0002-9947-98-01707-3
- PDF | Request permission
Abstract:
Let $G$ be the quasi-split unitary group $U(2,2)_{k’/k}$ of four variables attached to a quadratic extension $k’/k$ of number fields. An irreducible decomposition of the non-cuspidal discrete spectrum of $L^2$-automorphic forms of $G$ is obtained.References
- James Arthur, Eisenstein series and the trace formula, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 253–274. MR 546601
- James Arthur, Intertwining operators and residues. I. Weighted characters, J. Funct. Anal. 84 (1989), no. 1, 19–84. MR 999488, DOI 10.1016/0022-1236(89)90110-9
- Tetsuya Asai, On certain Dirichlet series associated with Hilbert modular forms and Rankin’s method, Math. Ann. 226 (1977), no. 1, 81–94. MR 429751, DOI 10.1007/BF01391220
- L. E. Dickson, All integers except $23$ and $239$ are sums of eight cubes, Bull. Amer. Math. Soc. 45 (1939), 588–591. MR 28, DOI 10.1090/S0002-9904-1939-07041-9
- Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, No. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917
- I. N. Bernšteĭn and A. V. Zelevinskiĭ, Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), no. 3(189), 5–70 (Russian). MR 0425030
- I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472. MR 579172, DOI 10.24033/asens.1333
- Yuval Z. Flicker, On distinguished representations, J. Reine Angew. Math. 418 (1991), 139–172. MR 1111204, DOI 10.1515/crll.1991.418.139
- Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of $\textrm {GL}(2)$ and $\textrm {GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542. MR 533066, DOI 10.24033/asens.1355
- David Goldberg, Some results on reducibility for unitary groups and local Asai $L$-functions, J. Reine Angew. Math. 448 (1994), 65–95. MR 1266747, DOI 10.1515/crll.1994.448.65
- Harish-Chandra, Automorphic forms on semisimple Lie groups, Lecture Notes in Mathematics, No. 62, Springer-Verlag, Berlin-New York, 1968. Notes by J. G. M. Mars. MR 0232893, DOI 10.1007/BFb0098434
- G. Harder, R. P. Langlands, and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal-Flächen, J. Reine Angew. Math. 366 (1986), 53–120 (German). MR 833013
- Tamotsu Ikeda, On the theory of Jacobi forms and Fourier-Jacobi coefficients of Eisenstein series, J. Math. Kyoto Univ. 34 (1994), no. 3, 615–636. MR 1295945, DOI 10.1215/kjm/1250518935
- Hervé Jacquet, Automorphic forms on $\textrm {GL}(2)$. Part II, Lecture Notes in Mathematics, Vol. 278, Springer-Verlag, Berlin-New York, 1972. MR 0562503, DOI 10.1007/BFb0058503
- Hervé Jacquet, On the residual spectrum of $\textrm {GL}(n)$, Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 185–208. MR 748508, DOI 10.1007/BFb0073148
- H. Jacquet and R. P. Langlands, Automorphic forms on $\textrm {GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654, DOI 10.1007/BFb0058988
- C. David Keys and Freydoon Shahidi, Artin $L$-functions and normalization of intertwining operators, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 1, 67–89. MR 944102, DOI 10.24033/asens.1551
- Henry H. Kim, The residual spectrum of $\textrm {Sp}_4$, Compositio Math. 99 (1995), no. 2, 129–151. MR 1351833
- Stephen S. Kudla, Splitting metaplectic covers of dual reductive pairs, Israel J. Math. 87 (1994), no. 1-3, 361–401. MR 1286835, DOI 10.1007/BF02773003
- Stephen S. Kudla, Stephen Rallis, and David Soudry, On the degree $5$ $L$-function for $\textrm {Sp}(2)$, Invent. Math. 107 (1992), no. 3, 483–541. MR 1150600, DOI 10.1007/BF01231900
- S. S. Kudla and W. J. Sweet, Jr., Degenerate principal series representations for $U(n,n)$, preprint.
- J.-P. Labesse and R. P. Langlands, $L$-indistinguishability for $\textrm {SL}(2)$, Canadian J. Math. 31 (1979), no. 4, 726–785. MR 540902, DOI 10.4153/CJM-1979-070-3
- Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 0579181, DOI 10.1007/BFb0079929
- Robert P. Langlands, Euler products, Yale Mathematical Monographs, vol. 1, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967. MR 0419366
- Soo Teck Lee, On some degenerate principal series representations of $\textrm {U}(n,n)$, J. Funct. Anal. 126 (1994), no. 2, 305–366. MR 1305072, DOI 10.1006/jfan.1994.1150
- Soo Teck Lee and Chen-bo Zhu, Degenerate principal series and local theta correspondence, Trans. Amer. Math. Soc. (to appear).
- L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
- J. Horn, Über eine hypergeometrische Funktion zweier Veränderlichen, Monatsh. Math. Phys. 47 (1939), 359–379 (German). MR 91, DOI 10.1007/BF01695508
- Colette Mœglin, Marie-France Vignéras, and Jean-Loup Waldspurger, Correspondances de Howe sur un corps $p$-adique, Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, Berlin, 1987 (French). MR 1041060, DOI 10.1007/BFb0082712
- Patrice Perrin, Représentations de Schrödinger, indice de Maslov et groupe metaplectique, Noncommutative harmonic analysis and Lie groups (Marseille, 1980) Lecture Notes in Math., vol. 880, Springer, Berlin-New York, 1981, pp. 370–407 (French). MR 644841
- S. Rallis, On the Howe duality conjecture, Compositio Math. 51 (1984), no. 3, 333–399. MR 743016
- S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089, DOI 10.1007/BF03046994
- Freydoon Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. MR 1070599, DOI 10.2307/1971524
- Freydoon Shahidi, Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), no. 4, 973–1007. MR 816396, DOI 10.1215/S0012-7094-85-05252-4
- Franz Rádl, Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen, Math. Z. 45 (1939), 429–446 (German). MR 82, DOI 10.1007/BF01580293
- Allan J. Silberger, Introduction to harmonic analysis on reductive $p$-adic groups, Mathematical Notes, vol. 23, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. Based on lectures by Harish-Chandra at the Institute for Advanced Study, 1971–1973. MR 544991
- J. A. Shalika and S. Tanaka, On an explicit construction of a certain class of automorphic forms, Amer. J. Math. 91 (1969), 1049–1076. MR 291087, DOI 10.2307/2373316
- David A. Vogan Jr., The unitary dual of $\textrm {GL}(n)$ over an Archimedean field, Invent. Math. 83 (1986), no. 3, 449–505. MR 827363, DOI 10.1007/BF01394418
- Nolan R. Wallach, Symplectic geometry and Fourier analysis, Lie Groups: History, Frontiers and Applications, Vol. V, Math Sci Press, Brookline, Mass., 1977. With an appendix on quantum mechanics by Robert Hermann. MR 0488148
- Yangbo Ye, Kloosterman integrals and base change for $\textrm {GL}(2)$, J. Reine Angew. Math. 400 (1989), 57–121. MR 1013725, DOI 10.1515/crll.1989.400.57
Bibliographic Information
- Takyua Kon-No
- Affiliation: Graduate School of Mathematics, Kyushu University, 33 Fukuoka 812, Japan
- Address at time of publication: Université Paris-Sud, Mathématiques, Bât. 425, F-91405 Orsay Cedex, France
- Email: takuya@math.kyushu-u.ac.jp
- Received by editor(s): April 20, 1995
- Received by editor(s) in revised form: November 7, 1995
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 1285-1358
- MSC (1991): Primary 11F70, 11F72; Secondary 11F27
- DOI: https://doi.org/10.1090/S0002-9947-98-01707-3
- MathSciNet review: 1370645
Dedicated: Dedicated to Professor H. Shimizu on his sixtieth birthday