Finite-dimensional representations of quantum affine algebras at roots of unity

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