  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.
