Infinitesimal operators for representations of complex Lie groups and Clebsch-Gordan coefficients for compact groups

Abstract

Explicit expressions are obtained for the infinitesimal operators of the degenerate representations of the groups SL(n,C), SO(n,C) and Sp(n,C) in a discrete basis. They are used to obtain the infinitesimal operators of unitary representations of the group K(X)K in a K basis, where K is one of the groups SU(n), SO(n), Sp(n). The subgroup K is diagonally embedded into K(X)K. Matrix elements (generalised Wigner d functions) of the degenerate representations of GL(n,C) and U(n)(X)U(n) are evaluated. Clebsch-Gordan series are derived for the tensor product of irreducible representations of K which are given by one non-zero integer. The infinitesimal operators are applied to obtain recurrence relations for the Clebsch-Gordan coefficients of this tensor product. It is remarkable that they connect Clebsch-Gordan coefficients corresponding to different resulting representations.