Differential structure on the dual of a compact lie group

In this paper we define difference operators and homogeneous Sobolev-type spaces on the dual of a compact Lie group. As an application and to show that this defines a relevant differential structure, we state and prove multiplier theorems of H\"ormander, Mihlin and Marcinkiewicz types together with the sharpness in the Sobolev exponent for the one of H\"ormander type.