Representations of SU ( 2 ) and Jacobi polynomials ∗

Literature for this section is for instance Sugiura [3, Ch. 1]. Let G be a group. Representations of G can be defined on any vector space (possibly infinite dimensional) over any field, but we will only consider representations on finite dimensional complex vector spaces. Let V be a finite dimensional complex vector space. Let GL(V ) be the set of all invertible linear transformations of V . This is a group under composition. If V has dimension n and if we choose a basis e1, . . . , en of V then the map x = x1e1 + · · · + xnen 7→ (x1, . . . , xn) : V → Cn is an isomorphism of vector spaces. There is a corresponding group isomorphism GL(V ) → GL(Cn) which sends each invertible linear transformation of V to the corresponding invertible matrix with respect to this basis. We denote GL(Cn) by GL(n,C): the group of all invertible complex n×n matrices. Here the group multiplication is by multiplication of matrices.