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Jacquet-Langlands correspondence and unitarizability. (Jacquet-Langlands et unitarisabilit√©.) (French) Zbl 1159.22005
The author proves some particular cases of the global Jacquet-Langlands correspondence and deduces that certain representations of an inner form of \(GL_n\) over a \(p\)-adic field are unitarizable. These results are based on studying a local Jacquet-Langlands correspondence for all smooth irreducible representations. This correspondence is characterized by the fact that it respects the classical Jacquet-Langlands correspondence and commutes with the parabolic induction functor.

MSC:
22E35 Analysis on \(p\)-adic Lie groups
11F70 Representation-theoretic methods; automorphic representations over local and global fields
20G05 Representation theory for linear algebraic groups
20G25 Linear algebraic groups over local fields and their integers
20G30 Linear algebraic groups over global fields and their integers
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