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Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations. With an appendix by Neven Grbac. (English) Zbl 1158.22018
In this paper the author proves a number of results about the local and the global Jacquet-Langlands correspondence. Local Jacquet-Langlands correspondence originally relates square-integrable representations of the general linear groups over a local \(p\)-adic field and square-integrable representations of its inner forms. The author (in this and previous papers) extends this correspondence (transfer) to all irreducible unitary representations of the general linear groups over local \(p\)-adic fields; the characterization in terms of characters of representations takes the same form as for the square-integrable representations. As a consequence, he establishes a number of results about the unitary dual of the general linear groups over \(p\)-adic division algebras (inner forms of the general linear groups over local fields) which he needs to establish the global Jacquet-Langlands correspondence. Let \(F\) be a global field of characteristic zero and let \(D\) be a central division algebra over \(F\) of dimension \(d^2.\) Then the global Jacquet-Langlands correspondence should relate the discrete series automorphic representations of the group of adèles of \(\text{GL}_{nd}(F)\) and the discrete series automorphic representations of the group of adèles of \(\text{GL}_n(D).\) The author establishes, based on his results on the local Jacquet-Langlands transfer, the global Jacquet–Langlands transfer. Using this transfer, he proves a number of important results on the discrete spectrum of the group of adèles of \(\text{GL}_n(D),\) namely the multiplicity one and the strong multiplicity one results. Also, he describes all the discrete spectrum of the aforementioned group using so-called basic cuspidal representations. The proofs of the global results rely on the trace formula.
In the appendix (by Neven Grbac) to this paper, using the Eisenstein series, the last step in the classification of the discrete spectrum is obtained: it is proved that the only cuspidal representations are actually the basic cuspidal ones.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
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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