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Some results on reducibility for unitary groups and local Asai \(L\)- functions. (English) Zbl 0815.11029
This paper is concerned with the local representation theory of the pair of groups of \(F\)-points of \(U(n,n, E/F)\) and \(\text{Res}_{E/F} (\text{GL}_ n)\) where \(E/F\) is a quadratic extension of local fields of characteristic zero and \(U(n,n, E/F)\) denotes the quasi-split unitary group associated with this extension. The author denotes by \(\Psi\) the adjoint representation of the \(L\)-group of \(\text{Res}_{E/F} (\text{GL}_ n)\) acting on the unipotent Levi component of a parabolic subgroup of \(\text{GL}_ n/E\) with reductive component \(\text{Res}_{E/F} (\text{GL}_ n)\). This generalizes a construction of Asai.
The author investigates in detail, using the representation theory of groups defined over local fields, the relationship between the analytic properties of the \(L\)-function \(L(s, \pi, \Psi)\), for an admissible representation of \(\text{Res}_{E/F} (\text{GL}_ n) (F) = \text{GL}_ n(E)\) and the properties of \(\pi\) under induction. The details of the results are too intricate to be summarized here. At the end of the paper the author also considers the case of \(U(n,n + 1, E/F)\).

MSC:
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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