Radon transformation on reductive symmetric spaces: support theorems

Abstract

In this thesis we introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and study some of their properties. In particular we obtain a generalization of Helgason's support theorem for the horospherical transform on a Riemannian symmetric space. A reductive symmetric space is a homogeneous space X=G/H for a reductive Lie group G of the Harish-Chandra class, where H is an open subgroup of the fixed-point subgroup for an involution sigma on G. By a sigma-parabolic subgroup of G we mean a parabolic subgroup P with sigma(P) opposite to P. Each such parabolic subgroup P has a unipotent radical N. We consider the Radon transform R mapping a function on X to a function on the space Xi of submanifolds xi = gN H in X. More precisely, if phi is a function on X, then R phi(xi) is defined by integrating phi over xi. This Radon transform, which is initially defined for compactly supported smooth functions, can be extended to a large class of distributions on X. If P is a minimal sigma- parabolic subgroup, then the submanifolds xi are called horospheres and the Radon transform is called the horospherical transform. If sigma is a Cartan involution, then X is a so-called Riemannian symmetric space of the non-compact type. In this case X has a natural Riemannian structure. Helgason proved in 1973 a support theorem for the horospherical transform on such a symmetric space. If phi is a function on X with support contained in a ball B in X, then obviously the horospherical transform of phi will be zero on all horospheres that do not intersect with B. The support theorem of Helgason states, surprisingly, that the converse of this implication is also true. In this thesis we generalize the support theorem of Helgason to a support theorem for the above described Radon transforms on a reductive symmetric space for distributions in a suitable class, containing the compactly supported ones. In the formulation and proof of the theorem certain convex cones play an important role. Although our support theorem applied to a Riemannian symmetric space seems at first sight to be weaker than the result by Helgason, the latter can be obtained from the first. Just as in the Riemannian case, the support theorem implies injectivity of the Radon transform