Quaternionic spherical harmonics and a sharp multiplier theorem on   quaternionic spheres
  A sharp $L^p$ spectral multiplier theorem of Mihlin--H\"ormander type is proved for a distinguished sub-Laplacian on quaternionic spheres. This is the first such result on compact sub-Riemannian manifolds where the horizontal space has corank greater than one. The proof hinges on the analysis of the quaternionic spherical harmonic decomposition, of which we present an elementary derivation. 
Julian Ahrens, Michael G. Cowling, Alessio Martini, Detlef M\"uller
math.AP math.FA