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We study invariant distributions on the tangent space to a symmetric space.We prove that an invariant distribution with the property that both its supportand the support of its Fourier transform are contained in the set ofnon-distinguished nilpotent orbits, must vanish. We deduce, using recentdevelopments in the theory of invariant distributions on symmetric spaces thatthe symmetric pair (GL(2n,R),Sp(2n,R)) is a Gelfand pair. More precisely, weshow 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 waspreviously proven for p-adic fields in [HR] for the field of complex numbers in[S].

Invariant distributions on non-distinguished nilpotent orbits with application to the Gelfand property of (GL(2n,R),Sp(2n,R))