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Let $L$ be a $p$-adic field $G = GL_n(L)$. Let $P$ be a standard parabolic subgroup with Levi decomposition $P = MU$, where $M \cong G_1\times \ldots \times G_r$, for $G_i \cong GL_{n_i}(L)$.

The following are classical:

1). If $\rho_i$ are discrete series representations of $G_i$, $\rho$ is the $M$ representation given by $(\rho_1 \otimes \ldots \otimes \rho_r)$, and $\pi$ is the image of $\rho$ under the normalized induction functor $I_P^G$, then $\pi$ is tempered.

2). All irreducible tempered representations arise in this way. To clarify, if $\pi$ is an irreducible tempered representation there is a parabolic subgroup $P$ with Levi subgroup $M$ and a discrete series representation $\rho$ of $M$ such that $\pi \cong I_P^G \rho$.

I've read this in multiple sources, and whenever they cite a source they point to Jacquet's "Generic Representations"; Rodier cites section 3 ("Tempered Representations") specifically. I must be missing something, as I don't see where in the paper these facts are proved.

What am I missing here?

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Do you assume that $\pi$ is irreducible ? You can find a proof in the book of David Renard, available on his webpage, p.343, VII.2.6.

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  • $\begingroup$ Looks good. Leaves open the question of "what am I missing in Jacquet's work?", but this is a secondary curiosity. Thanks! $\endgroup$ Dec 30, 2016 at 17:03

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