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Let $M$ be an odd-dimensional Euclidean space endowed with a contact 1-form$\alpha$. We investigate the space of symmetric contravariant tensor fields on$M$ as a module over the Lie algebra of contact vector fields, i.e. over theLie subalgebra made up by those vector fields that preserve the contactstructure. If we consider symmetric tensor fields with coefficients in tensordensities, the vertical cotangent lift of contact form $\alpha$ is a contactinvariant operator. We also extend the classical contact Hamiltonian to thespace of symmetric density valued tensor fields. This generalized Hamiltonianoperator on the symbol space is invariant with respect to the action of theprojective contact algebra $sp(2n+2)$. The preceding invariant operators leadto a decomposition of the symbol space (expect for some critical densityweights), which generalizes a splitting proposed by V. Ovsienko

Decomposition of symmetric tensor fields in the presence of a flat contact pro jective structure