The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces (Q1816503)

The Helgason Fourier transform on a noncompact Riemannian symmetric space \(C/K\) is generalized to the homogeneous vector bundles \(E^\tau\) \((\tau\in \widehat {K})\) over \(G/K\). The corresponding inversion formula is obtained by using the Plancherel formula on \(G\) and the subrepresentation theorem. For radial systems of sections of \(E^\tau\), the Helgason Fourier transform reduces to the (operator valued) spherical transform, defined with respect to the (operator valued) \(\tau\)-spherical functions on \(G\).