  We describe explicitly the canonical map $\chi:$ Spec $\ue(\a{g})\
\rightarrow \ $Spec $\ze$, where $\ue(\a{g})$ is a quantum loop algebra at an
odd root of unity $\ve$. Here $\ze$ is the center of $\ue(\a{g})$ and Spec $R$
stands for the set of all finite--dimensional irreducible representations of an
algebra $R$. We show that Spec $\ze$ is a Poisson proalgebraic group which is
essentially the group of points of $G$ over the regular adeles concentrated at
$0$ and $\infty$. Our main result is that the image under $\chi$ of Spec
$\ue(\a{g})$ is the subgroup of principal adeles.
