Multiplicity one theorem for (GL(n+1,R),GL(n,R))

Let F be either R or C. Consider the standard embedding GLn(F ) ↪→ GLn+1(F ) and the action of GLn(F ) on GLn+1(F ) by conjugation. In this paper we show that any GLn(F )-invariant distribution on GLn+1(F ) is invariant with respect to transposition. We show that this implies that for any irreducible admissible smooth Frechet representations π of GLn+1(F ) and τ of GLn(F ), dim HomGLn(F )(π, τ) ≤ 1. For p-adic fields those results were proven in [AGRS]. Mathematics Subject Classification (2000). 20G05, 22E45, 20C99, 46F10.