La formule des traces d’Arthur-Selberg. (Arthur-Selberg trace formula). (French) Zbl 0592.22011

Sémin. Bourbaki, 37e année, Vol. 1984/85, Exp. No. 636, Astérisque 133/134, 73-88 (1986).
[For the entire collection see Zbl 0577.00004.]
In a number of papers J. Arthur found a generalization of the classical Selberg trace formula. [Several important references are Duke Math. J. 45, 915-952 (1978; Zbl 0499.10032); Ann. Math., II. Ser. 114, 1- 74 (1981; Zbl 0495.22006); Am. J. Math. 104, 1243-1288 and 1289-1336 (1982; Zbl 0541.22010, Zbl 0562.22004); Can. J. Math. 38, 179-214 (1986).]
In a few words this generalization is the following. Let G be a reductive algebraic group defined over a number field F. The Selberg-Arthur trace formula is an identity \[ \sum_{{\mathfrak o}\in {\mathcal O}}J_{{\mathfrak o}}(f)=\sum_{\chi \in {\mathcal H}}J_{\chi}(f),\quad f\in C_ c^{\infty}(G({\mathbb{A}})) \] 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 paper under review is a creative survey of Arthur’s results and possible applications of the trace formula.
Reviewer: A.Venkov


22E40 Discrete subgroups of Lie groups
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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