Conformally Equivariant Quantization - a Complete Classification

Conformally equivariant quantization is a peculiar map between symbols ofreal weight d and differential operators acting on tensor densities, whose real weights aredesigned by l and l+d. The existence and uniqueness of such a map has been provedby Duval, Lecomte and Ovsienko for a generic weight d. Later, Silhan has determined thecritical values of d for which unique existence is lost, and conjectured that for those valuesof d existence is lost for a generic weight l. We fully determine the cases of existence anduniqueness of the conformally equivariant quantization in terms of the values of dand l.Namely, (i) unique existence is lost if and only if there is a nontrivial conformally invariantdifferential operator on the space of symbols of weight d, and (ii) in that case the conformallyequivariant quantization exists only for a finite number of l, corresponding to nontrivialconformally invariant differential operators on l-densities. The assertion (i) is proved in themore general context of IFFT (or AHS) equivariant quantization.