Let X₀ be the tangent space at eH of the reductive symmetric space G H, and let G₀ denote the group of affine transformations of X₀ generated by the translations and the natural action of H. We show that any joint eigenspace of the G₀-invariant differential operators on X₀ is scalarly irreducible under the action of G₀. This holds in particular for a Riemannian symmetric space of the non-compact type, where G₀ is the Cartan motion group.