Invariant distributions on non-distinguished nilpotent orbits with application to the gelfand property of (GL _2n(ℝ), Sp _2n(ℝ))

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 Fréchet representation (π,E) of (GL _2n(ℝ) the space of continuous functionals Hom _Sp2n(ℝ)(E,ℂ) is at most one dimensional. Such a result was previously proven for p-adic fields in M. J. Heumos and S. Rallis, Symplectic-Whittaker models for Gl _n , Pacific J. Math. 146 (1990), 247-279, and for ℂ in E. Sayag, (GL _2n(ℝ), Sp _2n(ℝ)) is a Gelfand pair, arXiv:0805.2625 [math.RT].