Normalization of intertwining operators and reducibility of representations induced from cuspidal ones; the case of \(p\)-adic classical groups. (Normalisation des opérateurs d’entrelacement et réductibilité des induites de cuspidales; le cas des groupes classiques \(p\)-adiques.) (French) Zbl 0956.22012

Let \(G(n)\) denote a symplectic or an orthogonal group of rank \(n\) over a non-archimedean local field \(F\). Let \(\pi\) be an irreducible cuspidal representation of \(G(n,F)\) and \(\rho\) an irreducible unitary cuspidal representation of \(\text{GL} (c,F)\). Then \(\text{GL} (c)\times G(n)\) is imbedded in \(G(n+c)\) as a Levi group of a maximal parabolic subgroup and we have the induced representation \(I(\rho,\pi,s) = \rho |\text{det}|^s \times \pi\) of \(\text{GL} (n+c,F)\). The subject of the article is the determination of the values of \(s\) for which this representation is reducible and the normalization of the intertwining operators from \(I(\rho,\pi,s)\) to \(I(\rho^{\ast},\pi,-s)\). A normalization is given under the assumption that \(\pi\) comes from a global representation which satisfies a certain functoriality condition with respect to a general linear group.


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