A Characterization of Differentiable Modules over Kac-Moody Groups

Abstract:

      For any integrable module (V, dπ) over a Kac-Moody algebra g′ we can lift it to a module (V,π) over the Kac-Moody group G associatedto g′ .The present paper characterizes such modules and shows that a G -module (V,π) is a such module lifted from an integrable g′ -module (V, π) if and only if the restriction of V toany root subgroup U_ａ is rational. By the characterization, we obtain aninteresting result which says that all finite dimensional modules over G(Q) are different iablefor the Chevalley group G(Q).