Invariant Differential Operators and the Radon Transform on the Horocycle Spaces

Abstract: We investigate the Radon transform for double fibrations of the horocycle spaces for the semisimple symmetric spaces with respect to the inclusion incidence relations. We present the inversion formula, support theorem and the range theorem by the invariant differential operators or the invariant system of differential operators constructed from the left action of the Pfaffian type elements in the universal enveloping algebra for the transformations group. In order to prove the range theorem, we make the explicit calculations of the Pfaffian type elements which lead to the calculations for the Harish-Chandra isomorphism of the central elements of the universal enveloping algebra. We deal with the Radon transform on the Schwartz space, the compactly supported smooth function spaces and the space of the sections of the line bundle. The range theorem on the space of the sections of the line bundle yields the range theorem for the Radon transform for double fibrations of compact homogeneous spaces which is not necessarily symmetric.