Radon transform on symmetric matrix domains

Let $\text{bbK}=\mathbb{R},\mathbb{C},\mathbb{H}$ be the field of real, complex or quaternionic numbers and $M_{p,q}(\text{bbK})$ the vector space of all $p\times q$-matrices\mathrm{.}Let$X$bethematrixunitballin$M_{n-r, r}(\bbK)$ consisting of contractive matrices. As a symmetric space, $X=G/K=O(n-r, r)/O(n-r)\times O®$,$U(n-r, r)/U(n-r)\times U®$andrespectively$Sp(n-r, r)/Sp(n-r)\times Sp®$\mathrm{.}Thematrixunitball$y_0$in$M_{r^\prime-r, r}$with$r^\prime \le n-1$isatotallygeodesicsubmanifoldof$X$andlet$Y$bethesetofall$G$-translationsofthesubmanifold$y_0$\mathrm{.}Theset$Y$ is then a manifold and an affine symmetric space. We consider the Radon transform $\mathcal{R}f(y)$ for functions $f\in C_0^{\mathrm{\infty }}(X)$ defined by integration of $f$ over the subset $y$, and the dual transform ${\mathcal{R}}^{t}F(x),x\in X$ for functions $F(y)$ on $Y$. We find inversion formulas by constructing explicit certain invariant differential operators.