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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
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