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A trace formula for coverings of connected reductive groups. III: Development of spectral closure. (La formule des traces pour les revêtements de groupes réductifs connexes. III: Le développement spectral fin.) (French) Zbl 1330.11038
This paper is the third part of the author’s impressive series [J. Reine Angew. Math. 686, 37–109 (2014; Zbl 1295.22027); Ann. Sci. Éc. Norm. Supér. (4) 45, No. 5, 787–859 (2012; Zbl 1330.11037); Math. Ann. 356, No. 3, 1029–1064 (2013) under review; Ann. Inst. Fourier 64, No. 6, 2379–2448 (2014; Zbl 1315.11041)], in which the invariant trace formula of J. Arthur [J. Am. Math. Soc. 1, No. 3, 501–554 (1988; Zbl 0667.10019)] is generalized to an $$m$$-fold central extension of $$G(\mathbb{A})$$, where $$G$$ is a connected reductive algebraic group defined over a number field $$F$$, and $$\mathbb{A}$$ is the ring of adèles of $$F$$. For the central extension it is assumed that the covering map splits over $$G(F)$$ and that the spherical Hecke algebras are commutative at all non-Archimedean places, which makes the study of automorphic forms possible.
This part of the series is concerned with the spectral side of the trace formula, which is a sum over cuspidal automorphic data, that is, Weyl group orbits of a semi-standard Levi subgroup and its cuspidal automorphic representation. It follows the work of J. Arthur [Am. J. Math. 104, 1243–1288 (1982; Zbl 0541.22010); Am. J. Math. 104, 1289–1336 (1982; Zbl 0562.22004)] in providing a fine expansion of the terms on the spectral side. The main result is a formula for the terms on spectral side for test functions that are finite for a fixed maximal compact subgroup.
Furthermore, as in [J. Arthur, J. Am. Math. Soc. 1, No. 3, 501–554 (1988; Zbl 0667.10019)], rearranging the terms on the spectral side by summing up the terms with the same norm of the imaginary part of the infinitesimal character and attached to the full group $$G$$ as a Levi factor in the fine expansion, one obtains the discrete part of the spectral side (of a fixed norm). The discrete part is expressed as a sum of traces of certain discrete spectrum representations of the central extension with certain coefficients. These coefficients are defined for representations of every semi-standard Levi subgroup using canonically normalized local weighted characters.

##### MSC:
 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 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
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##### References:
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