Some results on reducibility of parabolic induction for classical groups

Abstract

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⋊σ is reducible if there exists ρ in the supercuspidal support of π such that ρ⋊σ is reducible. In special cases we also give irreducibility criteria for π⋊σ when the above condition is not satisfied.