Radon transform on real, complex and quaternionic Grassmannians

Let $G_{n,k}(\bbK)$ be the Grassmannian manifold of $k$-dimensional $\bbK$-subspaces in $\bbK^n$ where $\bbK=\mathbb R, \mathbb C, \mathbb H$ is the field of real, complex or quaternionic numbers. For $1\le k < k^\prime \le n-1$ we define the Radon transform $(\mathcal R f)(\eta)$, $\eta \in G_{n,k^\prime}(\bbK)$, for functions $f(\xi)$ on $G_{n,k}(\bbK)$ as an integration over all $\xi \subset \eta$. When $k+k^\prime \le n$ we give an inversion formula in terms of the G\aa{}rding-Gindikin fractional integration and the Cayley type differential operator on the symmetric cone of positive $k\times k$ matrices over $\bbK$. This generalizes the recent results of Grinberg-Rubin for real Grassmannians.