Multiplicity one Conjectures

TL;DR

This paper investigates conjectures related to invariant distributions on GL(n+1) and their implications for multiplicity one theorems, providing proofs for small n and extending results to orthogonal and unitary groups.

Contribution

It proves the multiplicity one conjecture for certain cases and extends the results to orthogonal and unitary groups based on GL(n) cases.

Findings

Distributions invariant under GL(n) are conjectured to be transposition invariant.

Proof of the conjecture for n<9.

Extension of results to orthogonal and unitary groups.

Abstract

In the first part, in the local non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We conjecture that such distributions are invariant by transposition. This would imply multiplicity at most one for restrictions from GL(n+1) to GL(n). We reduce ourselves to distributions with "singular" support and then finish the proof for n< 9. In the second part we show that similar Theorems for orthogonal or unitary groups follow from the case of GL(n)