Pfaffian systems and Radon transforms on affine Grassmann manifolds (Q1407657)

The authors investigate Radon transforms on affine Grassmann manifolds. The paper is organized as follows: In \S1, the notations used are introduced and range theorems for integral transforms in terms of Helgason's duality in integral geometry are considered. In \S2, the Pfaffian-type elements \(W_I\) of the universal enveloping algebra \(u(\text{so}(n))\) are introduced, while in \S3 the Casimir, or central, elements in \(u(\text{so}(n))\) are expressed in terms of these Pfaffians. In \S4, the authors show how these Casimir elements serve to characterize the range of the transforms \(R_{p,q}\) for the corresponding compact Grassmannians \(G_{p,n}\). The analogous Pfaffian-type elements \(V_J\) and Casimir elements \(Q_{2k}\) of the universal enveloping algebra \(u(m(n))\) of the Euclidean group are constructed, The \S6 and \S7 comprise the main parts of the paper. The inversion formula for \(R_{(p,q)}\) and the range characterization theorem in terms of the aforementioned Pfaffian-type elements are obtained in these sections.