Conical Distributions on the Space of Flat Horocycles

Let G0 = K ⋉ p be the Cartan motion group associated with a noncompact semisimple Riemannian symmetric pair (G, K). Let a be a maximal abelian subspace of p and let p = a+ q be the corresponding orthogonal decomposition. A flat horocycle in p is a G0-translate of q. A conical distribution on the space Ξ0 of flat horocycles is an eigendistribution of the algebra D(Ξ0) of G0-invariant differential operators on Ξ0 which is invariant under the left action of the isotropy subgroup of G0 fixing q. We prove that the space of conical distributions belonging to each generic eigenspace of D(Ξ0) is one-dimensional, and we classify the set of all conical distributions on Ξ0 when G/K has rank one.