  Let F be a non-archimedean local field of characteristic zero. We consider
distributions on GL(n+1,F) which are invariant under the adjoint action of
GL(n,F). We prove that any such distribution is invariant with respect to
transposition. This implies that the restriction to GL(n) of any irreducible
smooth representation of GL(n+1) is multiplicity free.
  Our paper is based on the recent work [RS] of Steve Rallis and Gerard
Schiffmann where they made a remarkable progress on this problem.
  In [RS], they also show that our result implies multiplicity one theorem for
restrictions from the orthogonal group $O(V \oplus F)$ to $O(V)$.
