Some results on reducibility of parabolic induction for classical groups

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