On the cohomology of sℓ( m + 1, [formula omitted]) acting on differential operators and sℓ( m + 1, [formula omitted])-equivariant symbol

Abstract

Let D k λμ ( R m) denote the space of differential operators of order ≤ k from the space of λ-densities of R m into that of μ-densities and let S δ k ( R m) be the space of k- contravariant, symmetric tensor fields on R m valued in the δ-densities, δ = μ − λ. Denote by sℓ m + 1 the projective embedding of sℓ( m + 1, R ) as a subalgebra of the Lie algebra of vector fields of R m. One computes the cohomology of sℓ m + 1 with coefficients in the space of differential operators from S δ p ( R m) into S δ q ( R m). It is non vanishing only for some critical values of δ. For m = 1, these are the values pointed out by H. Gargoubi in a completely different context [6]. This allows to determine the condition under which the short exact sequence of sℓ m + 1 - modules0 → D λμ k − 1( R m) → D λμ k( R m) σ → S δ k → 0 is split (σ is the symbol map). From this one recovers and generalizes useful results about the structure of the sℓ m + 1 - moduleD λμ( R m) = ∪ kD λμ k( R m) [3, 5, 6, 8]. The cohomology of the latter is also computed.