Integral Geometry on Grassmann Manifolds and Calculus of Invariant Differential Operators

In this paper, we mainly deal with two problems in integral geometry, the range characterizations and construction of inversion formulas for Radon transforms on higher rank Grassmann manifolds. The results will be described explicitly in terms of invariant differential operators. We will also study the harmonic analysis on Grassmann manifolds, using the method of integral geometry. In particular, we will give eigenvalue formulas and radial part formulas for invariant differential operators.