A note on special orthogonal groups following Waldspurger

TL;DR

This paper verifies that the archimedean multiplicity one theorems for orthogonal groups also hold for special orthogonal groups, using Waldspurger's ingredients.

Contribution

It extends previous multiplicity one results from orthogonal groups to special orthogonal groups by leveraging Waldspurger's methods.

Findings

Multiplicity one theorems confirmed for special orthogonal groups

Results align with previous theorems for orthogonal groups

Uses Waldspurger's ingredients to establish the variant

Abstract

The purpose of this note is to verify that the archimedean multiplicity one theorems shown for orthogonal groups (as well as general linear and unitary groups) in a previous paper of the authors remain valid for special orthogonal groups. The necessary ingredients to establish this variant are due to Waldspurger.