The Helgason Fourier transform for homogeneous vector bundles over compact Riemannian symmetric spaces—the local theory

The Helgason Fourier transform on noncompact Riemannian symmetric spaces is generalized to the homogeneous vector bundles over the compact dual spaces . The scalar theory on was considered by Sherman (the local theory for of arbitrary rank, and the global theory for of rank one). In this paper we extend the local theory of Sherman to arbitrary homogeneous vector bundles on . For of rank one we also obtain a generalization of the Cartan-Helgason theorem valid for any K-type.