Irreducible components of L2 functions on hopf bundles

This paper provides an explicit decomposition of the L² function space on the unit sphereS^dn-1 for d = 1,2 and 4, into irreducible representations under the action of the Lie Groups K= SO(n) × SO(1)S(U(n) × U(1)), and Sp(n) × Sp(l), respectively. The decomposition is realized as the eigenspaces of the Laplacian acting on homogeneous polynomials over the reals, complex numbers and quaternions. For the quaternionic case, an additional differential operator that commutes with the Laplacian is used to find the decomposition.