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The Helgason Fourier transform on a noncompact Riemannian symmetric space G/K is generalized to the homogeneous vector bundles E (τ ∈ 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 , the Helgason Fourier transform reduces to the (operator valued) spherical transform, defined with respect to the (operator valued) τ-spherical functions on G.

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