In addition to what Carlo Beenakker stated, you can find a detailed introduction to Fourier Transforms, Spherical Harmonics and Wigner-D matrices in Chapter 3 of a 2017 PhD thesis at Bielefeld University linked here.
It focuses on applications for visual robot navigation (and therefore especially on real-valued implementations), but a brief summary of the general idea is given. For a purely math based focus, you can check the references.
