Radon transform on real, complex, and quaternionic Grassmannians

Let G(n,k)(K) be the Grassniannian manifold of k-diniensional K-subspaces in K-n, where K = R, C, H is thefield of real, complex, or quaternionic numbers. For 1 <= k < k' <= n - 1, we define the Radon transform (Rf)(eta), n epsilon G(n,k') (K), for functions f (xi) on G(n,k)(K) as an integration over all xi subset of eta. When k + k' <= n, we give an inversion formula in terms of the Garding-Gindikin fractional integration and the Cavley-type differential operator on the symmetric cone of positive (k x k)-matrices over K. This generalizes the recent results of Grinberg and Rubin [4] for real Grassmannians.