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

Abstract

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 . For U / K of rank one we also obtain a generalization of the Cartan-Helgason theorem valid for any K-type.