Multiplicity one theorem for (GL(n+1,R),GL(n,R))
  Let F be either R or C. Consider the standard embedding GL(n,F)<GL(n+1,F) and the action of GL(n,F) on GL(n+1,F) by conjugation.   In this paper we show that any GL(n,F)-invariant distribution on GL(n+1,F) is invariant with respect to transposition.   We show that this implies that for any irreducible admissible smooth Frechet representations $\pi$ of GL(n+1,F) and $\tau$ of GL(n,F),   $$dim Hom_{GL(n,F)}(\pi,\tau) \leq 1.$$   For p-adic fields those results were proven in [AGRS]. 
Avraham Aizenbud and Dmitry Gourevitch