The study of a particular non-linear dispersive partial differential equation usually requires a version of pseudo-differential calculus. In this talk, we aim to introduce a toolbox of coordinate-independent para-differential calculus defined on compact Lie groups. We will first briefly review previous approaches for pseudo-differential and para-differential calculus on compact manifolds, together with their applications to dispersive equations. Next, we will construct para-differential calculus on a compact Lie group using representation theory, emphasizing the role played by localization property and classical differential symbols. Finally, we will describe how this para-differential toolbox applies to the spherical capillary water waves equation, a non-local, quasi-linear dispersive differential equation defined on the 2-sphere.