Orbites unipotentes et poles d'ordre maximal de la fonction $\mu $ de Harish-Chandra

In a previous work, we have shown that a representation of a $p$-adic group obtained by (normalized) parabolic induction from an irreducible supercuspidal representation $\sigma$ of a Levi subgroup $M$ contains a subquotient which is square integrable, if and only if Harish-Chandra's $\mu$-function has a pole in $\sigma$ of order equal to the parabolic rank of $M$. The aim of the present article is to interpret this result in terms of Langlands' functoriality principle.