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 GLn(F) on GLn+1(F) by conjugation. We show that any GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We prove that this implies that for any irreducible admissible smooth Frechet representations π of GLn+1(F) and \(\tau\) of GLn(F), $${\rm dim\,Hom}_{{\rm GL}_{n}(F)}(\pi, \tau) \leq 1$$ . For p-adic fields those results were proven in [AGRS].