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 $G/K$ is generalized to the homogeneous vector bundles over the compact dual spaces $U/K$. The scalar theory on $U/K$ was considered by Sherman (the local theory for $U/K$ of arbitrary rank, and the global theory for $U/K$ of rank one). In this paper we extend the local theory of Sherman to arbitrary homogeneous vector bundles on $U/K$.