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 a of sigma classical group, we show that the (normalized) parabolic induction pi (sic) sigma is reducible if there exists rho in the supercuspidal support of pi such that rho (sic) sigma is reducible. In special cases we also give irreducibility criteria for pi (sic) sigma when the above condition is not satisfied.