Multiplicity One Theorems and Invariant Distributions

TL;DR

This thesis investigates the properties of invariant distributions and proves that the pair (GL(n+1,F), GL(n,F)) forms a strong Gelfand pair, establishing a key multiplicity one result in representation theory.

Contribution

It introduces tools for working with invariant distributions and proves the strong Gelfand property for the pair (GL(n+1,F), GL(n,F)), a significant advancement in understanding multiplicity one theorems.

Findings

Proved that (GL(n+1,F), GL(n,F)) is a strong Gelfand pair.

Established that the multiplicity of Hom spaces between certain representations is at most one.

Developed methods to analyze invariant distributions in the context of Gelfand pairs.

Abstract

This is my PhD thesis submitted to the Weizmann Institute of Science. It is based on the papers [AG08c], [AG08d], [AGRS07], [AGS08], [AGS09], [Aiz08] and [SZ08]. This thesis includes an introduction to Gelfand pairs and invariant distributions, a list of tools to work with invariant distributions oriented towards proving Gelfand property and a proof that the pair (GL(n+1,F),GL(n,F)) is a strong Gelfand pair. Namely, we prove that if $\pi$ is an irreducible admissible smooth representation of GL(n+1,F) and $\rho$ is an irreducible admissible smooth representation of GL(n,F) then $$dim Hom_{GL(n,F)}(\pi,\rho)\leq 1.$$