Multiplicity one Theorems

TL;DR

This paper proves that certain invariant distributions on GL(n+1) are symmetric, leading to multiplicity one results for restrictions of representations, with similar theorems for orthogonal and unitary groups.

Contribution

It establishes new multiplicity one theorems for representations of GL(n+1) and related groups by analyzing invariant distributions and their symmetries.

Findings

Distributions invariant under GL(n) are also invariant under transposition.

Admissible irreducible representations of GL(n+1) restrict to GL(n) with multiplicity one.

Similar results hold for orthogonal and unitary groups.

Abstract

In the local, characteristic 0, non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant by transposition. This implies that an admissible irreducible representation of GL(n+1), when restricted to GL(n) decomposes with multiplicity one. Similar Theorems are obtained for orthogonal or unitary groups.