Article

Selecta Mathematica

, Volume 15, Issue 2, pp 271-294

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

  • Avraham AizenbudAffiliated withFaculty of Mathematics and Computer Science, The Weizmann Institute of Science Email author 
  • , Dmitry GourevitchAffiliated withFaculty of Mathematics and Computer Science, The Weizmann Institute of Science

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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].

Mathematics Subject Classification (2000).

20G05 22E45 20C99 46F10

Keywords.

Multiplicity one Gelfand pair invariant distribution coisotropic subvariety