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

Abstract

We describe explicitly the canonical map χ : \chi : Spec U ε ( g ~ ) → U_{\varepsilon }(\tilde {\mathfrak {g}}) \rightarrow Spec Z ε Z_{\varepsilon } , where U ε ( g ~ ) U_{\varepsilon } (\tilde {\mathfrak {g}}) is a quantum loop algebra at an odd root of unity ε \varepsilon . Here Z ε Z_{\varepsilon } is the center of U ε ( g ~ ) U_{\varepsilon }(\tilde {\mathfrak {g}}) and Spec R R stands for the set of all finite–dimensional irreducible representations of an algebra R R . We show that Spec Z ε Z_{\varepsilon } is a Poisson proalgebraic group which is essentially the group of points of G G over the regular adeles concentrated at 0 0 and ∞ \infty . Our main result is that the image under χ \chi of Spec U ε ( g ~ ) U_{\varepsilon }(\tilde {\mathfrak {g}}) is the subgroup of principal adeles.