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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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