Eigenfunction Expansions of Functions Describing Systems with Symmetries

Physical systems with symmetries are described by functions containing kinematical and dynamical parts. We consider the case when kinematical symmetries are described by a noncompact semisimple real Lie group $G$. Then separation of kinematical parts in the functions is fulfilled by means of harmonic analysis related to the group $G$. This separation depends on choice of a coordinate system on the space where a physical system exists. In the paper we review how coordinate systems can be chosen and how the corresponding harmonic analysis can be done. In the first part we consider in detail the case when $G$ is the de Sitter group $SO_0(1,4)$. In the second part we show how the corresponding theory can be developed for any noncompact semisimple real Lie group.