The Coxeter element and the branching law for the finite subgroups of SU ( 2 )

0.1. Let Γ be a finite subgroup of SU(2). The question we will deal with in this paper is how an arbitrary (unitary) irreducible representation of SU(2) decomposes under the action of Γ. The theory of McKay assigns to Γ a complex simple Lie algebra g of type A−D−E. The assignment is such that if Γ̃ is the unitary dual of Γ we may parameterize Γ̃ by the nodes (or vertices) of the extended Coxeter-Dynkin diagram of g. Let l = rank g and let I = {1, . . . , l}. Let Iext = I ∪ {0}. The nodes may be identified with a set of simple roots of the affine Kac-Moody Lie algebra associated to g and are indexed by Iext. We can then write Γ = {γi}, i ∈ Iext. Let Π = {αi}, i ∈ I, be the set of simple roots of g itself. One has γ0 is the trivial 1 dimensional representation of Γ and, for i ∈ I,