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The Residual Spectrum of G2

Published online by Cambridge University Press:  20 November 2018

Henry H. Kim
Henry H. Kim*
Affiliation:
Dept. of Math., Southern Illinois University, Carbondale, IL 62901 e-mail: henrykim@math.siu.edu
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Abstract

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We completely determine the residual spectrum of the split exceptional group of type G2, thus completing the work of Langlands and Moeglin-Waldspurger on the part of the residual spectrum attached to the trivial character of the maximal torus. We also give the Arthur parameters for the residual spectrum coming from Borel subgroups. The interpretation in terms of Arthur parameters explains the “bizarre” multiplicity condition in Moeglin-Waldspurger's work. It is related to the fact that the component group of the Arthur parameter is non-abelian.

Keywords

11F7022E55
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 6 , 01 December 1996 , pp. 1245 - 1272
Copyright
Copyright © Canadian Mathematical Society 1996

References

[A1] Arthur, J., On some problems suggested by the trace formula, In: Lie group representations, II, Lecture Notes in Math. 1041, Springer-Verlag, 1984, 1–49.Google Scholar
[A2] Arthur, J., Unipotent Automorphic Representations: Conjectures, In: Orbites unipotentes et representations II. Groupesp-adiques et reels, Asterisque 171-172(1989), 13–71.Google Scholar
[A3] Arthur, J., Eisenstein Series and the Trace Formula, Proceedings of Symposia in Pure Mathematics, I, 33(1979), 253–274.Google Scholar
[B-V] Barbasch, D. and Vogan, D., Jr., Unipotent representations of complex semisimple groups, Ann. of Math. 121(1985), 41–110.Google Scholar
[Bu-G-H] Bump, D., Ginzburg, D. and Hoffstein, J., The Symmetric Cube, 1995, preprint.Google Scholar
[Ca] Carter, R. W., Finite Groups of Lie Type, Conjugacy Classes and Complex Characters, John Wiley and Sons, 1985.Google Scholar
[G-S] Gelbart, S. and Shahidi, F., Analytic Properties of Automorphic L-functions, Perspectives in Mathematics 6, Academic Press, 1988.Google Scholar
[Go] Goldstein, L., Analytic Number Theory, Prentice-Hall, 1971.Google Scholar
[Ik] Ikeda, T., On the location of poles of the triple L-functions, Comp. Math. 83(1992), 187–237.Google Scholar
[Ja] Jacquet, H., On the residual spectrum of GL(n), In: Lie Group Representations II, Lecture Notes in Math. 1041, Springer-Verlag, 1983, 185–208.Google Scholar
[K-S] Keys, C. D. and Shahidi, F., Artin L-functions and normalization of intertwining operators, Ann. Scient. Ec. Norm. Sup. 21(1988), 67–89.Google Scholar
[Ki] Kim, H., The Residual Spectrum Sp4, Comp. Math. (2)99(1995), 129–151.Google Scholar
[Ki-S] Kim, H. and Shahidi, F., Quadratic unipotent Arthur parameters and residual spectrum of Sp2n , Amer. J. Math. (2)118, 401–425.Google Scholar
[L1] Langlands, R. P., Representations of abelian algebraic groups, Notes, Yale University, 1968.Google Scholar
[L2] Langlands, R. P., On the classification of irreducible representations of real algebraic groups, In: Representation Theory and Harmonic Analysis on Semisimple Lie Groups, (eds. Sally, Paul J. and Vogan, David A.), AMS Math. Surveys and Monographs 31(1989), 101-170.Google Scholar
[L3] Langlands, R. P., Euler Products, Yale University Press, 1971.Google Scholar
[L4] Langlands, R. P., On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math. 544, Springer-Verlag, 1976.Google Scholar
[Li-Sc] Li, J. S. and Schwermer, J., Constructions of automorphie forms and related cohomology classes for arithmetic subgroups of G2, Comp. Math. 87(1993), 45–78.Google Scholar
[M1] Moeglin, C., Représentations unipotentes et formes automorphes de carreintegrable, Forum mathematician 6(1994), 651–744.Google Scholar
[M2] Moeglin, C., Orbites unipotentes et spectre discret non ramifie, Le cas desgroupes classiques deploy e, Comp. Math. 77(1991), 1–54.Google Scholar
[M3] Moeglin, C., Une conjecture sur le spectre résiduel des groupes classiques, 1994, preprint.Google Scholar
[M4] Moeglin, C., Letter, 1994.Google Scholar
[M5] Moeglin, C., Cuspidal quadratic unipotent representations, 1994, preprint.Google Scholar
[M-W1] Moeglin, C. and Waldspurger, J. L., Spectral decomposition and Eisenstein series, Une Paraphrase de l'Écriture, Cambridge University Press, 1995.Google Scholar
[M-W2] Moeglin, C. and Waldspurger, J. L., Le Spectre Résiduel de GL(n), Ann. Scient. Éc. Norm. Sup. 22(1989), 605–674.Google Scholar
[R] Rodier, F., Decomposition de la serie principale des groupes redutifs p-adiques, In: Non-commutative harmonic analysis, Lecture Notes in Math. 880, Springer-Verlag, 1981, 408–424.Google Scholar
[S1] Shahidi, F., On certain L-functions, Amer. J.Math. 103(1981), 297–356.Google Scholar
[S2] Shahidi, F., Whittaker Models for Real Groups, Duke Math. J. (1)47(1980), 99–125.Google Scholar
[S3] Shahidi, F., On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. of Math. 127(1988), 547–584.Google Scholar
[S4] Shahidi, F., Third symmetric power L-functions for GL(2) Comp. Math. 70(1989), 245–273.Google Scholar
[Sp] Spaltenstein, N., Classes Unipotentes et Sous-groupes de Borel, Lecture Notes in Mathematics 946, Springer-Verlag, 1982.Google Scholar
[Wi] Winarsky, N., Reducibility of principal series representations of p-adic Chevalley groups, Amer. J. Math. (5)100(1978), 941–956.Google Scholar