Noncompact Lie-algebraic approach to the unitary representations of SU∼(1,1): Role of the confluent hypergeometric equation

The continuous unitary irreducible representations of the covering group SU∼ (1,1) of SU(1,1) are studied using the self-adjoint representations of the Lie algebra in the case that one of the noncompact generators is diagonal. The study can be carried out for all the representations simultaneously and is shown to reduce to a study of the self-adjointness of the compact element of the Lie algebra, which in this basis turns out to be the confluent hypergeometric operator. Several basic results, such as the classification of the representations, and a formula for the transformation coefficients from the compact to the noncompact basis which is valid for all representations, emerge quite simply.