A proof of the multiplicity one conjecture for GL(n) in GL(n+1)

TL;DR

This paper proves the multiplicity one conjecture for restrictions of irreducible representations from GL(n+1) to GL(n) over non-archimedean fields, confirming multiplicity-free restrictions using invariant distribution techniques.

Contribution

It establishes that invariant distributions under GL(n,F) on GL(n+1,F) are symmetric, leading to a proof of the multiplicity one conjecture for GL(n) restrictions.

Findings

Any invariant distribution is symmetric under transposition.

Restrictions of irreducible representations are multiplicity free.

Implication for multiplicity one theorems in related orthogonal groups.

Abstract

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 $O(V)$.