Conformally covariant bi-differential operators for differential forms

TL;DR

This paper generalizes classical bi-differential operators known as Rankin-Cohen brackets to higher dimensions and differential forms, ensuring conformal covariance under the action of the conformal group.

Contribution

It introduces conformally covariant bi-differential operators for differential forms on -dimensional spaces, extending classical operators to a broader geometric setting.

Findings

Constructed new bi-differential operators for differential forms.

Proved covariance properties under conformal group actions.

Extended classical operators to higher-dimensional conformal geometry.

Abstract

The classical Rankin-Cohen brackets are bi-differential operators from $C^\infty(\mathbb R)\times C^\infty(\mathbb R)$ into $ C^\infty(\mathbb R)$. They are covariant for the (diagonal) action of ${\rm SL}(2,\mathbb R)$ through principal series representations. We construct generalizations of these operators, replacing $\mathbb R$ by $\mathbb R^n,$ the group ${\rm SL}(2,\mathbb R)$ by the group ${\rm SO}_0(1,n+1)$ viewed as the conformal group of $\mathbb R^n,$ and functions by differential forms.