Let F be a local field of characteristic 0. We consider
distributions on GL(n+1,F) which are invariant under the adjoint action of
GL(n,F). We prove that such distributions are invariant under
transposition. This implies that an irreducible representation of
GL(n+1,F), when restricted to GL(n,F) "decomposes" with multiplicity one.
Such property of a group and a subgroup is called strong Gelfand property.
It is used in representation theory and automorphic forms. This property
was introduced by Gelfand in the 50s for compact groups. However, for
non-compact groups it is much more difficult to establish.
For our pair (GL(n+1,F),GL(n,F)) it was proven in 2007 in [AGRS] for
non-Archimedean F, and in 2008 in [AG] and [SZ] for Archimedean F. In this
lecture we will present a uniform for both cases. This
proof is based on the above papers and an additional new tool. If time
permits we will discuss similar theorems that hold for orthogonal and
unitary groups.
[AG] A. Aizenbud, D. Gourevitch: "Multiplicity one theorem for
(GL(n+1,R),GL(n,R))", arXiv:0808.2729v1 [math.RT]
[AGRS] A. Aizenbud, D. Gourevitch, S. Rallis, G. Schiffmann:
"Multiplicity One Theorems", arXiv:0709.4215v1 [math.RT], to appear in the
Annals of Mathematics.
[SZ] B. Sun and C.-B. Zhu" "Multiplicity one theorems: the archimedean
case", preprint available at
http://www.math.nus.edu.sg/~matzhucb/Multiplicity_One.pdf