Projectively equivariant symbol calculus

Abstract

The spaces of linear differential operators on ${\mathbb{R}}^n$ acting on tensor densities of degree $\lambda$ and the space of functions on $T^{\ast }{\mathbb{R}}^n$ 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 $s{l}_{n+1}$-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 $s{l}_{n+1}$-equivariant symbol map to study the $\Vect(M)$-modules of linear differential operators acting on tensor densities, for an arbitrary manifold $M$.

23 pages, LaTeX This article is a revised version of the electronic preprint dg-ga/9611006