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A trace formula for reductive groups. II : applications of a truncation operator

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A trace formula for reductive groups. II: applications of a truncation operator COMPOSITIO MATHEMATICA JAMESARTHUR A trace formula for reductive groups. II: applications of a truncation operator Compositio Mathematica, tome 40, no 1 (1980), p. 87-121. <http://www.numdam.org/item?id=CM_1980__40_1_87_0> © Foundation Compositio Mathematica, 1980, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 87 A TRACE FORMULA FOR REDUCTIVE GROUPS II: APPLICATIONS OF A TRUNCATION OPERATOR James Arthur* COMPOSITIO MATHEMATICA, Vol. 40, Fasc. 1, 1980, pag. 87-121 @ 1980 Sijthoff &#x26; Noordhoff International Publishers - Alphen aan den Rijn Printed in the Netherlands 1. A truncation operator ............. ’ 89 2. Integrability of k X (x, f ) ............ 98 3. The operator M J;( 1T) ............. 107 4. Evaluation in a special case ........... 113 5. Conclusion ................. 120 Introduction This paper, as promised in the introduction to [1(c)], contains an identity which is valid for any reductive group G over Q, and which generalizes the Selberg trace formula for anisotropic G. We have already shown that a certain sum of distributions on O(A)I, indexed by équivalence classes in G(Q), equals the intégral of the function The main task of this paper is to show that the integral may be taken inside the sum over x. There does not seem to be any easy way to do this. We are forced to proceed indirectly by first defining and studying a truncation operator AT on functions on G(G)BG(A)’. Recall that k x T(

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