Conformally invariant differential operators on tensor densities

Abstract

Let ${\mathcal{F}}_{\lambda }$ be the space of tensor densities on ${\mathbf{R}}^n$ of degree $\lambda$ (or, equivalently, of conformal densities of degree $-\lambda n$) considered as a module over the Lie algebra $so(p+1,q+1)$. We classify $so(p+1,q+1)$-invariant bilinear differential operators from ${\mathcal{F}}_{\lambda }\otimes {\mathcal{F}}_{\mu }$ to~${\mathcal{F}}_{\nu }$. The classification of linear $so(p+1,q+1)$-invariant differential operators from ${\mathcal{F}}_{\lambda }$ to ${\mathcal{F}}_{\mu }$ already known in the literature is obtained in a different manner.

11 pages, LaTeX