In Stephen Kudla's paper "The local Langlands correspondence: the non-Archimedean case", we find the following result about square-integrable (mod center) representations and Langlands quotients $Q(\Delta)$.
Proposition 1.2.3 (I. N. Bernstein, [48, Proposition 11]). Every square integrable representation $\pi$ of $G$ has the form $Q(\Delta)$, where $$ \Delta=(\sigma, \sigma(1), \sigma(2), \ldots, \sigma(r-1)) $$ with $\sigma\left(\frac{r-1}{2}\right)$ unitary.
The reference [48] is Rodier's Bourbaki seminar talk, which states
PROPOSITION 11.- Toute représentation de carré intégrable de $\mathrm{G}_{\mathrm{n}}$ est de la forme $\mathrm{L}(\Delta)$ pour un segment $\Delta=[\rho, \rho(\mathrm{r})]$ où $\rho$ est une représentation cuspidale irréductible telle que $\rho\left(\frac{r}{2}\right)$ soit unitaire.
Notice the difference between $r-1$ and $r$, and between $Q(\Delta)$ and $\mathrm{L}(\Delta)$, in their notations.
I find this proposition false in the simplest case where $G=\mathrm{GL}_1(\mathbb{Q}_p)$ in which case we must have $r-1=0$ in Kudla's notation (or $r=0$ in Rodier's notation). But then every character of $G$ is square-integrable (modulo center) as $G$ is abelian, but not every smooth character $\chi: \mathbb{Q}_p^\times \to \mathbb{C}^\times$ is unitary (i.e., with values in $S^1$) : we are free to send $p$ to any nonzero complex number.
Question. I wonder what the correct statement of this proposition is. Could any expert give a correction or a idea of the proof? Is it fine if we restrict ourselves to $G = \mathrm{GL}_n$ for $n>1$? In fact, Rodier's usage of $r$, as opposed to $r-1$, seems to suggest this.
Thank you in advance!
