Let $G$ be a connected reductive group over a finite extension $F$ of $\mathbb{Q}_p$. Let $P = MN$ be a Levi decomposition of a maximal parabolic subgroup of $G$, and $\sigma$ an irreducible unitary supercuspidal representation of $M(F)$. One can then consider the representation Ind$_{P(F)}^{G(F)}\sigma$ (normalized parabolic induction). This induced representation is known to be either irreducible or of length two. The question of when it is irreducible turns out to be (conjecturally) related to local $L$-functions, and also to poles of a family of so called intertwining operators.

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