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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 $\operatorname{Ind}_P^G\sigma$?

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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.

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  • $\begingroup$ If $\pi$ is a discrete series representation, can we say more about its multiplicity? And is there a similar theory of $R$-groups for general parabolics (not necessarily minimal)? $\endgroup$
    – youknowwho
    Commented Sep 26, 2023 at 18:07

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