The trace formula for the coverings of connected reductive groups. IV: Invariant distributions. (La formule des traces pour les revêtements de groupes réductifs connexes. IV: Distributions invariantes.) (French. English summary) Zbl 1315.11041
This paper is the final 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; Zbl 1330.11038)], in which the invariant trace formula of Arthur 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 \(F\). The final formula in such generality is proved subject to the technical hypothesis that the invariant Paley-Wiener theorem holds at real places. For some important cases, such as central extensions of the general linear group \(\mathrm{GL}(n,\mathbb{R})\) and the metaplectic group, that is, the two-fold central extension of the symplectic group \(\mathrm{Sp}(2n,\mathbb{R})\), it is checked that the hypothesis holds.
The importance of the invariant trace formula is in that it is a step towards the stabilization of the trace formula, and the stable trace formula plays the key role in the endoscopic classification of automorphic representations [J. Arthur, The endoscopic classification of representations. Orthogonal and symplectic groups. Colloquium Publications 61. Providence, RI: American Mathematical Society (2013; Zbl 1310.22014); C. P. Mok, Mem. Am. Math. Soc. 1108 (2015; Zbl 1316.22018)]. The generalization to finite central extensions of reductive groups is important, because they appear naturally in the arithmetic considerations, for instance, in modular forms of half-integral weight, theta correspondence, and multiple Dirichlet series.
The invariant trace formula for connected reductive groups over a number field is finally established in [J. Arthur, J. Am. Math. Soc. 1, No. 2, 323–383 (1988; Zbl 0682.10021), J. Am. Math. Soc. 1, No. 3, 501–554 (1988; Zbl 0667.10019)]. However, more recently, J. Arthur [J. Inst. Math. Jussieu 1, No. 2, 175–277 (2002; Zbl 1040.11038)] provided another approach to obtain the invariant trace formula with a view toward the stable trace formula. This latter approach is followed in the present paper and the serious obstacles of working with a central extension are overcome.
The importance of the invariant trace formula is in that it is a step towards the stabilization of the trace formula, and the stable trace formula plays the key role in the endoscopic classification of automorphic representations [J. Arthur, The endoscopic classification of representations. Orthogonal and symplectic groups. Colloquium Publications 61. Providence, RI: American Mathematical Society (2013; Zbl 1310.22014); C. P. Mok, Mem. Am. Math. Soc. 1108 (2015; Zbl 1316.22018)]. The generalization to finite central extensions of reductive groups is important, because they appear naturally in the arithmetic considerations, for instance, in modular forms of half-integral weight, theta correspondence, and multiple Dirichlet series.
The invariant trace formula for connected reductive groups over a number field is finally established in [J. Arthur, J. Am. Math. Soc. 1, No. 2, 323–383 (1988; Zbl 0682.10021), J. Am. Math. Soc. 1, No. 3, 501–554 (1988; Zbl 0667.10019)]. However, more recently, J. Arthur [J. Inst. Math. Jussieu 1, No. 2, 175–277 (2002; Zbl 1040.11038)] provided another approach to obtain the invariant trace formula with a view toward the stable trace formula. This latter approach is followed in the present paper and the serious obstacles of working with a central extension are overcome.
Reviewer: Neven Grbac (Rijeka)
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 |
Citations:
Zbl 1295.22027; Zbl 0682.10021; Zbl 0667.10019; Zbl 1040.11038; Zbl 1330.11037; Zbl 1330.11038; Zbl 1310.22014; Zbl 1316.22018References:
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