New Approach to Matrix Elements and Clebsch-Gordan Coefficients for Compact and Non-Compact Groups

Abstract

Linear representations of Lie groups are useful in both physics and mathematics. The most important applicable aspects of representations theory are matrix elements and Clebsch-Gordan (CG) coefficients of group representations. New results of representation theory allow us to give a new approach to matrix elements and CG coefficients. At first, matrix elements and CG coefficients of unitary irreducible representations were studied separately for compact and non-compact semisimple Lie groups and for each series of representations (the principal unitary series, the supplementary series, the discrete series and so on). These studies can now be linked together. This link is realized by the principal nonunitary series representations (“analytic continuation” in the continuous parameters of the principal unitary series representations) of a semisimple noncompact Lie group. The point is that every completely irreducible representation of such a group is contained in some representation of the principal nonunitary series. On the other hand, matrix elements of the principal nonunitary series representations at fixed group element are entire analytic functions of continuous parameters defining representations. Therefore, if matrix elements of representations of one of the continuous series (for example, of the principal unitary series) are known then those of representations of other series can be obtained by an appropriate analytic Continuation. However, most of the unitary representations which are contained in the principal nonunitary series representations are nonunitary. Therefore, it is necessary to evaluate the matrices of unitarization of unitarizable representations. These matrices bear a simple relationship with intertwining operators for the principal nonunitary series representations. For this reason an evaluation of matrices of intertwining operators is an important task. Different intertwining operators are linked by CG coefficients of the tensor product of the principal nonunitary series representations with finite dimensional representations of the group.