Multiplicity one theorem for $({\rm GL}_{n+1}({\mathbb{R}}), {\rm GL} _ {n} ({ \mathbb{R}}))$

Abstract.

Let F be either ${\mathbb{R}}$ or ${\mathbb{C}}$. Consider the standard embedding ${\rm GL}_n(F) \hookrightarrow {\rm GL}_{n+1}(F)$ and the action of GL _n (F) on GL_n+1(F) by conjugation. We show that any GL _n (F)-invariant distribution on GL _n+1 (F) is invariant with respect to transposition. We prove that this implies that for any irreducible admissible smooth Fréchet representations π of GL _n+1 (F) and $\tau$ of GL _n (F),

$${\rm dim\,Hom}_{{\rm GL}_{n}(F)}(\pi, \tau) \leq 1$$

. For p-adic fields those results were proven in [AGRS].