Let $G$ be a $p$-adic reductive group and $\pi$ an irreducible non-supercuspidal representation. Then there exist a parbaolic subgroup $P=MN$ and a supercuspidal representation of $M$ such that $\pi$ appears as a subrepresentation of $\operatorname{Ind}\_P^G\sigma$, namely $\sigma$ is the supercuspidal support of $\pi$.


Now, is it known that $\pi$ appears with mupltiplicity one in $\operatorname{Ind}\_P^G\sigma$?



On a theorem of Bernstein-Zelevinsky regarding supercuspidal resentations
Higher multiplicities can occur. See Keys, *L-indistinguishability and R-groups for quasisplit groups: unitary groups in even dimension*, Ann. Sci. ENS, 4th series, vol 20, no. 1, 1987, pp. 31-64.


Given a minimal parabolic subgroup $B=TU$ and a unitary character $\lambda$ of $T$, the multiplicities of the components occurring in $\text{Ind}\_B^G \lambda$ are controlled by the so-called $R$-group of $\lambda$. In particular, each component corresponds to an irreducible representation of $R$, and its multiplicity is the dimension of that representation. This paper gives examples where $R$ is nonabelian, in which case some multiplicities will be greater than one.


I suspect that for some groups $G$, the associated $R$-groups are all known to be abelian, but someone more knowledgeable can chime in on that.


