Reducibility of some induced representations of split classical p-adic groups

In this paper we study reducibility of representations of split classical p-adic groups induced from self-contragredient supercuspidal representations of general linear groups. For a supercuspidal representation associated via Howe's construction to an admissible character, we show that in many cases Shahidi's criterion for reducibility of the induced representation reduces to a simple condition on the admissible character.