Some results on reducibility of parabolic induction for classical groups

Given a (complex, smooth) irreducible representation $π$ of the general linear group over a non-archimedean local field and an irreducible supercuspidal representation $σ$ of a classical group, we show that the (normalized) parabolic induction $π\rtimesσ$ is reducible if there exists $ρ$ in the supercuspidal support of $π$ such that $ρ\rtimesσ$ is reducible. In special cases we also give irreducibility criteria for $π\rtimesσ$ when the above condition is not satisfied.

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