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.