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?



How does Jacquet's "Generic Representations" classify tempered representations?
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.


