Eigenfunction expansions of functions describing systems with symmetries (Q2473453)

Every symmetry of a physical system is described by some group \(G\). In general a function describing such a system is characterized by a part which is defined by the symmetry and a part which is related with a concrete system. As an example, if we are dealing with a system of differential equations containing a symmetry group, then one can often reduce it to consider a simpler system with no symmetry. To split a describing function of a system into two parts, one for the symmetry and one for the concrete system, consists in fact in separating the system into kinematic and dynamical parts. The authors of this paper deal with this separation. First, a suitable set of variables can be chosen in such a way that some of them correspond to the symmetries and the remaining ones correspond to dynamics of the system. The functions describing a system are considered with respect to this choice of variables and in the framework of the harmonic analysis of the kinematical part, the functions are expanded into basis functions which are common eigenfunctions of a collection of self-adjoint operators that are determined by the symmetry group. There exists a one-to-one correspondence between the following collections: a collection of kinematical variables; a chain of subgroups of the symmetry group \(G\); a collection of self-adjoint operators that are, as a rule, Casimir operators of the group \(G\) and of members of the chain of subgroups. In the article under review the description is reviewed of such triples in the case that the symmetry group \(G\) is a simple noncompact Lie group. In particular, in the first part the case is considered in detail when \(G\) is the de Sitter group \(SO_0 (1,4)\) and in the second part it is shown how the corresponding theory can be developed for any noncompact semisimple Lie group. Substantial references for this review are two books: \textit{I. I. Kachurik} and \textit{A. U. Klimyk}, Numerical methods in the theory of group representations (Russian). Kiev: Vishcha Shkola (1986; Zbl 0649.22011); \textit{A. U. Klimyk} and \textit{N. Ya. Vilenkin}, Representation of Lie groups and special functions. Volume 2: Class I representations, special functions, and integral transforms. Mathematics and Its Applications. Soviet Series. 74. Dordrecht: Kluwer Academic Publishers (1993; Zbl 0809.22001).