Conformally Equivariant Quantization - a Complete Classification

Conformally equivariant quantization is a peculiar map between symbols of
real weight d and differential operators acting on tensor densities, whose real weights are
designed by l and l+d. The existence and uniqueness of such a map has been proved
by Duval, Lecomte and Ovsienko for a generic weight d. Later, Silhan has determined the
critical values of d for which unique existence is lost, and conjectured that for those values
of d existence is lost for a generic weight l. We fully determine the cases of existence and
uniqueness of the conformally equivariant quantization in terms of the values of d
and l.

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 d, and (ii) in that case the conformally
equivariant quantization exists only for a finite number of l, corresponding to nontrivial
conformally invariant differential operators on l-densities. The assertion (i) is proved in the
more general context of IFFT (or AHS) equivariant quantization.