Unitary representations of three dimensional Lie groups revisited: An   approach via harmonic functions
  Harmonic functions of the three dimensional Lie groups defined on certain manifolds related to the Lie groups themselves and carrying all their unitary representations are explicitly constructed. The realisations of these Lie groups are shown to be related with each other by either natural operations as real forms or In\"on\"u-Wigner contractions. 
R. Campoamor-Stursberg and M. Rausch de Traubenberg
math-ph hep-th math.MP math.RT