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

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