Towards projectively equivariant quantizations
by
Pierre B. A. Lecomte
Université de Liège

Let us say that a quantization on a smooth manifold M is a bijection Q: S --> D between the space D of differential operators (acting on functions or on tensor densities of M) and the space S of their symbols that maps each P in S onto an operator Q(P) whose symbol is P. (Recall that S is the space of smooth functions on the cotangent bundle of M that are polynomial on the fibers.) We denote the space of these quantization by Q(M). Note that the reciprocal of a quantization is called a symbol or a symbol map.

It is well known that there is no natural quantization: there is no Q in Q(M) such that L_XQ=0 for all vector field X of M, L denoting the Lie derivative. On IR^m, it is however possible to build quantizations that are invariant under some Lie subalgebras of the Lie algebras of vector fields. This is clearly the case for the algebra of affine vector fields. It has also been shown to be the case for the projective embedding sl_m+1 of sl(m+1, IR) and for some embedding of the so(p+1, q+1), p+q=m, the quantization being then unique. Recently, for second order operators, with the help of a covariant derivation Ñ, Bouarroudj has constructed a bijection Q_Ñ : S² --> D² that preserves the symbol, that depends only of the projective class of Ñ and that reduces to the restriction of the above mentionned equivariant quantization if M=IR^m and Ñ is the canonical flat derivation. A similar construction has been obtained by Duval and Ovsienko for the conformal case.

Our aim in the talk is to study the problem of the existence of a natural projectively equivariant quantization map, that is a map Q : C_M --> Q(M), where C(M) is the space of covariant derivations with vanishing torsion of M, such that

(1) Q_Ñ only depends on the projective class of Ñ in C_M
(2) L_XQ=0 for all vector field X of M

As sl_m+1 is the space of infinitesimal automorphism of the projective structure of IR^m associated to its canonical flat derivation Ñ, these conditions imply that, for that derivation, the quantization Q_Ñ coincides with the unique sl_m+1-equivariant quantization of IR^m.

We will introduce the different cohomologies concerned by this problem, explain how to use them recover the result of Bouarroudj and, hopefully, to construct natural projectively equiveriqnt map Q.

Date received: March 26, 2001