  The spaces of linear differential operators on ${\mathbb{R}}^n$ acting on
tensor densities of degree $\lambda$ and the space of functions on
$T^*{\mathbb{R}}^n$ which are polynomial on the fibers are not isomorphic as
modules over the Lie algebra $\Vect({\mathbb{R}}^n)$ of vector fields on
${\mathbb{R}}^n$. However, these modules are isomorphic as
$sl(n+1,{\mathbb{R}})$-modules where $sl(n+1,{\mathbb{R}})\subset
\Vect({\mathbb{R}}^n)$ is the Lie algebra of infinitesimal projective
transformations. In addition, such an $sl_{n+1}$-equivariant bijection is
unique (up to normalization). This leads to a notion of projectively
equivariant quantization and symbol calculus for a manifold endowed with a
(flat) projective structure. We apply the $sl_{n+1}$-equivariant symbol map to
study the $\Vect(M)$-modules of linear differential operators acting on tensor
densities, for an arbitrary manifold $M$.
