Conformally covariant bi-differential operators for differential forms

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 $\mathrm{SL}(2,\mathbb{R})$ through principal series representations. We construct generalizations of these operators, replacing $\mathbb{R}$ by ${\mathbb{R}}^n,$ the group $\mathrm{SL}(2,\mathbb{R})$ by the group ${\mathrm{SO}}_0(1,n+1)$ viewed as the conformal group of ${\mathbb{R}}^n,$ and functions by differential forms.

23 pages