Radon Transform on Symmetric Matrix Domains

Let X be the matrix unit ball in Mn−k,k(K) consisting of contractive matrices where K = R, C, H. The domain X is a realization of the symmetric space G/K with G = U(n− k, k; K). The matrix ball yo of lower dimension in Mk′−k,k with k ′ ≤ n is a totally geodesic submanifold of X and let Y be the manifold of all G-translations of the submanifold y0. We consider the Radon transform from functions on X to functions on Y and we obtain an inversion formula. For that purpose we prove some Bernstein-Sato type formula for certain distributions which turn out to be closely related to Berezin transform.