Conformally Equivariant Quantization - a Complete Classification

Abstract

Conformally equivariant quantization is a peculiar map between symbols of real weight $\delta$ and differential operators acting on tensor densities, whose real weights are designed by $\lambda$ and $\lambda +\delta$. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight $\delta$. Later, Silhan has determined the critical values of $\delta$ for which unique existence is lost, and conjectured that for those values of $\delta$ existence is lost for a generic weight $\lambda$. We fully determine the cases of existence and uniqueness of the conformally equivariant quantization in terms of the values of $\delta$ and $\lambda$. Namely, (i) unique existence is lost if and only if there is a nontrivial conformally invariant differential operator on the space of symbols of weight $\delta$, and (ii) in that case the conformally equivariant quantization exists only for a finite number of $\lambda$, corresponding to nontrivial conformally invariant differential operators on $\lambda$-densities. The assertion (i) is proved in the more general context of IFFT (or AHS) equivariant quantization.

References in corpus (4)

• Conformally invariant quantization -- towards complete classification
• Singular conformally invariant trilinear forms and covariant differential operators on the sphere
• Invariant Bilinear Differential Pairings on Parabolic Geometries
• Invariant Differential Pairings

Cited by in corpus (5)

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• On osp(p+1,q+1|2r)-equivariant quantizations
• Quantization and injective submodules of differential operator modules
• Singular conformally invariant trilinear forms and generalized Rankin Cohen operators
• Conformally equivariant quantization for spinning particles