Selecta Mathematica

, Volume 15, Issue 2, pp 271–294

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

Authors

    • Faculty of Mathematics and Computer ScienceThe Weizmann Institute of Science
  • Dmitry Gourevitch
    • Faculty of Mathematics and Computer ScienceThe Weizmann Institute of Science
Article

DOI: 10.1007/s00029-009-0544-7

Cite this article as:
Aizenbud, A. & Gourevitch, D. Sel. Math. New Ser. (2009) 15: 271. doi:10.1007/s00029-009-0544-7

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 Fréchet 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].

Mathematics Subject Classification (2000).

20G0522E4520C9946F10

Keywords.

Multiplicity oneGelfand pairinvariant distributioncoisotropic subvariety

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009