Harmonic Analysis in Vector Bundles Associated to the Rotation and Spin Groups

Let V be any vector bundle over the sphere S n which is associated to the principal bundle of oriented orthonormal frames, or to that of spin frames. We give an explicit formula for the spectrum, with multiplicity, of the Bochner Laplacian r r on V , where r is the Riemannian or Riemannian spin connection. In the case of tensor bundles, we also give the spectrum of the Lichnerowicz Laplacian. The eeect of natural rst-order diierential operators on the corresponding eigenspaces is investigated, and all results are illustrated in a detailed treatment of some examples: the bundles of diierential forms, spinors, trace-free symmetric tensors, and algebraic Weyl tensors.