## The invariant trace formula. II: Global theory.(English)Zbl 0667.10019

[For part I, cf. ibid., No.2, 323-383 (1988)]
This paper is one in the author’s monumental series on the Selberg trace formula for arbitrary reductive groups. In it the ‘spectral’ and ‘geometric’ sides of the invariant form of the trace formula are analyzed further into elementary distributions. These are rather difficult to define in general and remain still somewhat inaccessible. Nevertheless they can be computed in many simplified cases in which the Selberg trace formula becomes a very effective tool. The crucial fact proved in this paper is that all the distributions appearing in the Selberg trace formula are supported on characters which is proved by a long and subtle induction argument.
Reviewer: S.J.Patterson

### MSC:

 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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### References:

 [1] James G. Arthur, A trace formula for reductive groups. I. Terms associated to classes in \?(\?), Duke Math. J. 45 (1978), no. 4, 911 – 952. · Zbl 0499.10032 [2] James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. · Zbl 0682.10022 [3] Joseph N. Bernstein, \?-invariant distributions on \?\?(\?) and the classification of unitary representations of \?\?(\?) (non-Archimedean case), Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 50 – 102. [4] J. N. Bernstein, Le ”centre” de Bernstein, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 1 – 32 (French). Edited by P. Deligne. · Zbl 0599.22016 [5] J. Bernstein, P. Deligne, and D. Kazhdan, Trace Paley-Wiener theorem for reductive \?-adic groups, J. Analyse Math. 47 (1986), 180 – 192. · Zbl 0634.22011 [6] L. Clozel and P. Delorme, Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs, Invent. Math. 77 (1984), no. 3, 427 – 453 (French). · Zbl 0584.22005 [7] L. Clozel, J. P. Labesse, and R. P. Langlands, Morning seminar on the trace formula, Lecture Notes, Institute for Advanced Study, Princeton, N. J., 1984. [8] P. Deligne, D. Kazhdan, and M. F. Vignéras, Représentations des algèbres centrales simples $$p$$-adic, Représentations des Groupes Réductifs sur un Corps Local, Hermann, Paris, 1984, pp. 33-117. · Zbl 0583.22009 [9] Harish-Chandra, Spherical functions on a semisimple Lie group. I, Amer. J. Math. 80 (1958), 241 – 310. · Zbl 0093.12801 [10] David Kazhdan, Cuspidal geometry of \?-adic groups, J. Analyse Math. 47 (1986), 1 – 36. · Zbl 0634.22009 [11] Robert E. Kottwitz, Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), no. 3, 365 – 399. · Zbl 0577.10028 [12] Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. · Zbl 0332.10018 [13] M. Raïs, Action de certains groupes dans des espaces de fonctions \?^{\infty }, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), Springer, Berlin, 1975, pp. 147 – 150. Lecture Notes in Math., Vol. 466. [14] J. D. Rogawski, Trace Paley-Wiener theorem in the twisted case, Trans. Amer. Math. Soc. 309 (1988), no. 1, 215 – 229. · Zbl 0663.22011 [15] David A. Vogan Jr., The unitary dual of \?\?(\?) over an Archimedean field, Invent. Math. 83 (1986), no. 3, 449 – 505. · Zbl 0598.22008
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