Conformally Equivariant Quantization - a Complete Classification

Abstract

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