  We prove that (GL_{2n}(C),Sp_{2n}(C)) is a Gelfand pair. More precisely, we
show that for an irreducible smooth admissible Frechet representation (\pi,E)
of GL_{2n}(C) the space of continuous functionals Hom_{Sp_{2n}(\cc)}(E,C) is at
most one dimensional. For this we show that any distribution on GL_{2n}(C)
invariant with respect to the double action Sp_{2n}(C) \times Sp_{2n}(C) is
transposition invariant. Such a result was previously proven for p-adic fields
by M. Heumos and S. Rallis.
