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On a family of distributions obtained from orbits. (English) Zbl 0593.22015

The paper under review is one from the sequence of the remarkable papers of the author the final aim of which is a derivation of the effectively determined invariant trace formula for general reductive groups G. The Selberg-Arthur trace formula is an identity \[ \sum_{{\mathfrak o}\in {\mathcal O}}J_{{\mathfrak o}}(f)=\sum_{\chi \in {\mathcal X}}J_{\chi}(f),\quad f\in C_ c^{\infty}(G({\mathbb{A}})) \] of distributions. The terms on the right are parametrized by ”cuspidal automorphic data”, and are defined in terms of Eisenstein series. The terms on the left are parametrized by semisimple conjugacy classes and are defined in terms of related G(\({\mathbb{A}})\) orbits.
The object of this paper is to evaluate the terms on the left. In his previous papers the author evaluated \(J_{{\mathfrak o}}(f)\) in two special cases: (1) ”\({\mathfrak o}''\) corresponds to a regular semisimple conjugacy class; (2)”\({\mathfrak o}''\) corresponds to \(\{\) \(1\}\). [See Duke Math. J. 45, 915-952 (1978; Zbl 0499.10032); Can. J. Math. 37, 1237-1274 (1985; Zbl 0589.22016).] Now the author reduces the general case of \(J_{{\mathfrak o}}\) to these special two.
Reviewer: A.Venkov

MSC:

22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11R56 Adèle rings and groups
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