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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.

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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