Vanishing of certain equivariant distributions on spherical spaces

We prove vanishing of distributions on a split real reductive group which change according to a non-degenerate character under the left action of the unipotent radical of the Borel subgroup, and are equivariant under the right action of a spherical subgroup, and eigen with respect to the center of the universal enveloping algebra of the Lie algebra of G. This is a generalization of a result by Shalika, that concerned the group case. Shalika's result was crucial in the proof of his multiplicity one theorem. We view our result as a step in the study of multiplicities of quasi-regular representations on spherical varieties. As an application we prove non-vanishing of spherical Bessel functions.