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

We describe explicitly the canonical map χ : Spec U_ε(g̃) → Spec Z_ε, where U_ε(g̃) is a quantum loop algebra at an odd root of unity ε. Here Z_ε is the center of U_ε(g̃) and Spec R stands for the set of all finite- dimensional irreducible representations of an algebra R. We show that Spec Z_ε is a Poisson proalgebraic group which is essentially the group of points of G over the regular adeles concentrated at 0 and ∞. Our main result is that the image under χ of Spec U_ε(g̃) is the subgroup of principal adeles.