Radon transformation on reductive symmetric spaces: Support theorems

We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and derive support theorems for these transforms. A reductive symmetric space is a homogeneous space 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 σ on G . Let P be a parabolic subgroup such that σ ( P ) is opposite to P and let N P be the unipotent radical of P . For a compactly supported smooth function ϕ on G / H , we define R P ( ϕ ) ( g ) to be the integral of N P ∋ n ↦ ϕ ( g n ⋅ H ) over N P . The Radon transform R P thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions. For these transforms we derive support theorems in which the support of ϕ is (partially) characterized in terms of the support of R P ϕ . The proof is based on the relation between the Radon transform and the Fourier transform on G / H , and a Paley–Wiener-shift type argument. Our results generalize the support theorem of Helgason for the Radon transform on a Riemannian symmetric space.