is the space ${{\mathcal{S}}}_{\d}(M)$ of fiberwise polynomials on $T^*M$ with values in $\d$-densities, where $\d=\mu-\l.$ In general, the space ${{\mathcal{D}}}_{\l,\mu}(M)$ is not isomorphic as a ${\mathrm{Diff}}(M)$-module to the space ${{\mathcal{S}}}_{\d}(M)$ (cf. [@do; @lo2]). However, we are interested in the following two cases:\
(i) If $M:={\mathbb{R}}^n$ is endowed with a flat projective structure (i.e. local action of the group ${\mathrm{SL}}_{n+1}({\mathbb{R}})$ by linear fractional transformations) there exists an isomorphism between ${{\mathcal{D}}}_{\l,\mu}({\mathbb{R}}^n)$ and ${{\mathcal{S}}}_{\d}({\mathbb{R}}^n),$ for $\d$ generic, intertwining the action of ${\mathrm{SL}}_{n+1}({\mathbb{R}})$ (cf. [@lo2]). The multi-dimensional “projective” Schwarzian derivative was defined in [@b2; @bo2] as an obstruction to extend this isomorphism to the full group ${\mathrm{Diff}}({\mathbb{R}}^n).$\
(ii) If $M:={\mathbb{R}}^n$ is endowed with a flat conformal structure (i.e. local action of the conformal group ${\mathrm{O}}(p+1,q+1),$ where $p+q=n$), there exists an isomorphism between ${{\mathcal{D}}}_{\l,\mu}({\mathbb{R}}^n)$ and ${{\mathcal{S}}}_{\d}({\mathbb{R}}^n),$ for $\d$ generic, intertwining the action of ${\mathrm{O}}(p+1,q+1)$ (cf. [@dlo; @do]). In this paper we introduce the multi-dimensional “conformal” Schwarzian derivative in this context. Recall that in the one-dimensional case these two notions coincide in the sense that the conformal Lie algebra ${\mathrm o}(2,1)$ is isomorphic to the projective Lie algebra ${\mathrm{sl}}_2({\mathbb{R}}).$

Differential operators and symbols
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Let $(M,\g)$ be a pseudo-Riemannian manifold of dimension $n.$ We denote by 