Multiplicity one theorem for (GL(n+1,R),GL(n,R))

TL;DR

This paper proves a multiplicity one theorem for the restriction of representations from GL(n+1,R) to GL(n,R), showing that invariant distributions are symmetric and that the Hom space dimension is at most one.

Contribution

It establishes a new multiplicity one result for real groups, extending prior p-adic results to the real case, and demonstrates invariance of distributions under transposition.

Findings

Invariant distributions on GL(n+1,R) are symmetric under transposition.

Dimension of Hom spaces between certain representations is at most one.

Results extend known p-adic theorems to real groups.

Abstract

Let F be either R or C. Consider the standard embedding GL(n,F)<GL(n+1,F) and the action of GL(n,F) on GL(n+1,F) by conjugation. In this paper we show that any GL(n,F)-invariant distribution on GL(n+1,F) is invariant with respect to transposition. We show that this implies that for any irreducible admissible smooth Frechet representations $\pi$ of GL(n+1,F) and $\tau$ of GL(n,F), $$dim Hom_{GL(n,F)}(\pi,\tau) \leq 1.$$ For p-adic fields those results were proven in [AGRS].