## The invariant trace formula. I: Local theory.(English)Zbl 0682.10021

This paper is one of a pair which is devoted to the deduction of an invariant form of the author’s general Selberg trace formula. The second part has already been reviewed [ibid. 1, No.3, 501-554 (1988; Zbl 0667.10019)]. The argument involves a complicated induction and the two papers have to be read together. Two families of distributions $$I_ M(\pi,*)$$ and $$I_ M(\gamma,*)$$ (M a reductive group, $$\pi$$ a unitary admissible representation of M and $$\gamma$$ a conjugacy class in M($${\mathbb{Q}}))$$ appear in the final version of the trace formula. The induction makes use of the dependence in all variables, including M. The global argument is described in Part II of the paper. In this part the local theory is developed. In particular the definitions of the distributions $$I_ M(\pi,*)$$ and $$I_ M(\gamma,*)$$ are given and the analytic properties are established.
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 22E50 Representations of Lie and linear algebraic groups over local fields

Zbl 0667.10019
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### References:

 [1] James Arthur, The trace formula in invariant form, Ann. of Math. (2) 114 (1981), no. 1, 1 – 74. · Zbl 0495.22006 [2] L. Clozel, Base change for \?\?(\?), Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 791 – 797. [3] 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 [4] Arne Brøndsted, An introduction to convex polytopes, Graduate Texts in Mathematics, vol. 90, Springer-Verlag, New York-Berlin, 1983. · Zbl 0509.52001 [5] Laurent Clozel and Patrick Delorme, Sur le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs réels, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 11, 331 – 333 (French, with English summary). · Zbl 0593.22009 [6] L. Clozel, J.-P. Labesse, and R. Langlands, Morning seminar on the trace formula, Lecture Notes, Institute for Advanced Study, Princeton Univ. [7] 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 [8] J. D. Rogawski, Trace Paley-Wiener theorem in the twisted case, Trans. Amer. Math. Soc. 309 (1988), no. 1, 215 – 229. · Zbl 0663.22011 [9] D. Shelstad, \?-indistinguishability for real groups, Math. Ann. 259 (1982), no. 3, 385 – 430. · Zbl 0506.22014
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