Let $\G$ be a connected reductive group over a finite extension
$F$ of $\mathbb{Q}_p$. Let $\P = \M \N$ 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 $\text{Ind}_{\P(F)}^{\G(F)} \sigma$ (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.
This calls for:
(a) computing residues of certain families of intertwining operators;
and
(b) interpreting these residues suitably.
There is an approach pioneered by Freydoon Shahidi to implement such a
programme, which was developed further by him as well as by David
Goldberg, Steven Spallone, Wen-Wei Li and Xiaoxiang Yu,
in several cases (i.e., for various choices of $\G$ and $\P$). We will
discuss (b) above for some of the cases where only (a) was known
previously, and also (a) for some new cases.