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A quantization over a manifold can be seen as a way to construct adifferential operator with prescribed principal symbol. The quantization map ismoreover required to be a linear bijection. It is known that there is ingeneral no natural quantization procedure. However, considering manifoldsendowed with additional structures, such as projective or pseudo-conformalstructures, one can seek for quantizations that depend on this additionalstructure and that are natural if the dependence with respect to the structureis taken into account.  The question of existence of such a quantization was addressed in a series ofpapers in the context of projective geometry, using the framework ofThomas-Whitehead connections (see Bordemann, Hansoul and Fox). Recently, werecovered these existence results, using the theory of Cartan projectiveconnections.  In the present work, we show that our method can be adapted topseudo-conformal geometry to yield the so-called natural and conformallyinvariant quantization for arbitrary symbols, still outside some criticalsituations. Our method is general enough to analyze the problem of invariantquantizations in the context of manifolds with irreducible parabolicgeometries.

Existence of natural and conformally invariant quantizations of arbitrary symbols