Arthur’s multiplicity formula for certain inner forms of special orthogonal and symplectic groups. (English) Zbl 1430.11074
In [The endoscopic classification of representations. Orthogonal and symplectic groups. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1310.22014)] J. Arthur announced a formula for the multiplicity of cuspidal representations in \(L^2(G(F)\backslash G({\mathbb A}))\), where \(G\) is an orthogonal group or a symplectic group over a number field \(F\) with adele ring \(\mathbb A\). In this formula, the multiplicity is given in explicit representation theoretic terms. At the time, Arthur had to work around the fact that the canonical absolute transfer factors had not been fully constructed yet, a problem which has been solved in the meantime by T. Kaletha [Invent. Math. 213, No. 1, 271–369 (2018; Zbl 1415.11079)]. Building on that, the author of the present paper is able to give a proof of the multiplicity formula under some technical restrictions. The proof relies on the stabilisation of the trace formula.
Reviewer: Anton Deitmar (Tübingen)
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
| 11R39 | Langlands-Weil conjectures, nonabelian class field theory |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
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