Radon transforms on higher rank grassmannians

We define a Radon transform R from functions Gr(k, n), the Grassmannian of projective A:-planes in CPⁿ to functions on Gr(l, n). If ∊ C^∞(Gr(k, n)) and L ∊ Gr(l, n), then Rf(L) is the integral of f(H) over all k planes H which lie in L. If R^t is the dual transform, we show under suitable assumptions on k and l that R^tR is invertible by a polynomial in the Casimir operators of U(n + 1), the group of isometries CPⁿ. We also treat the real and quaternionic cases. Finally, we indicate some possible variations and generalizations to flag manifolds.