On the kernel of the maximal flat Radon transform on symmetric spaces of compact type

Let $M$ be a Riemannian globally symmetric space of compact type, $M'$ its set of maximal flat totally geodesic tori, and $\mathrm{ad}(M)$ its adjoint space. We show that the kernel of the maximal flat Radon transform $\tau:L^2(M) \rightarrow L^2(M')$ is precisely the orthogonal complement of the image of the pullback map $L^2(\mathrm{ad}(M))\rightarrow L^2(M)$. In particular, we show that the maximal flat Radon transform is injective if and only if $M$ coincides with its adjoint space.