Projectively equivariant symbol calculus

TL;DR

This paper develops a projectively equivariant symbol calculus for differential operators on manifolds with flat projective structures, establishing unique $sl(n+1)$-equivariant bijections and applying them to study modules of differential operators.

Contribution

It introduces a novel projectively equivariant symbol calculus and quantization method for manifolds with flat projective structures, extending module isomorphisms to arbitrary manifolds.

Findings

Established $sl(n+1)$-equivariant bijections between differential operators and symbols.

Applied the equivariant symbol map to modules on arbitrary manifolds.

Proved the uniqueness of the equivariant bijection up to normalization.

Abstract

The spaces of linear differential operators on ${\mathbb{R}}^n$ acting on tensor densities of degree $\lambda$ and the space of functions on $T^*{\mathbb{R}}^n$ 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 $M$.