The author gives a survey of recent work concerning the mapping properties of joint harmonic projection operators \[ \pi_p:L^2(S^{dn-1})\to V^p \] mapping the space of square integrable functions noncomplex and quaternionic spheres onto the eigenspaces of the Laplace-Beltrami operator and of a suitably defined subLaplacian. In particular, the similarities and differences are discussed between the real ( \(V^p={\mathcal H^l}\) is the space of spherical harmonics), the complex (\(V^p={\mathcal V^{l,l'}}\) is the space of complex spherical harmonics) and the quaternionic (\(V^p\) is the joint eigenspace of the Laplace Beltrami operator) framework.
For the entire collection see [Zbl 1367.35008].