  We study invariant distributions on the tangent space to a symmetric space.
We prove that an invariant distribution with the property that both its support
and the support of its Fourier transform are contained in the set of
non-distinguished nilpotent orbits, must vanish. We deduce, using recent
developments in the theory of invariant distributions on symmetric spaces that
the symmetric pair (GL(2n,R),Sp(2n,R)) is a Gelfand pair. More precisely, we
show that for any irreducible smooth admissible Frechet representation
$(\pi,E)$ of GL(2n,R) the space of continuous functionals
$Hom_{Sp_{2n}(R)}(E,C)$ is at most one dimensional. Such a result was
previously proven for p-adic fields in [HR] for the field of complex numbers in
[S].
