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

Abstract. Let M be an odd-dimensional Euclidean space endowed with a contact 1-form α. We investigate the space of symmetric con-travariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vec-tor fields that preserve the contact structure. If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form α is a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symmetric density valued tensor fields. This generalized Hamiltonian operator on the space of symbols is invariant with respect to the action of the projective contact algebra sp(2n+2). These two operators lead to a decomposition of the space of symbols (except for some critical density weights), which generalizes a splitting proposed by V. Ovsienko in [18]. 1