We study the space of bilinear differential operators on weighted densities as a module over sl(2, R). We introduce the corresponding space of symbols and we prove the existence and the uniqueness of canonical projective equivariant symbol and quantization maps.

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We consider the supercircle [Formula: see text] equipped with the standard contact structure. The conformal Lie superalgebra [Formula: see text] acts on [Formula: see text] as the Lie superalgebra of contact vector fields; it contains the M[Formula: see text]bius superalgebra [Formula: see text]. We study the space of linear differential operators on weighted densities as a module over [Formula: see text]. We introduce the canonical isomorphism between this space and the corresponding space of symbols. This result allows us to give, in contrast to the classical setting, a classification of the [Formula: see text]-modules [Formula: see text] of linear differential operators of order [Formula: see text] acting on the superspaces of weighted densities. This work is the simplest superization of a result by Gargoubi and Ovsienko [Modules of differential operators on the real line, Funct. Anal. Appl. 35(1) (2001) 13–18.]

Let F n λ be the space of tensor densities of degree λ ∈ C on the supercircle S 1|n . We consider the space D n,k λ,μ of k-th order linear differential operators from F n λ to F n μ as a module over the superalgebra K(n) of contact vector fields on S 1|n and we compute the superalgebra of endomrphisms on D n,k λ,μ commuting with the aff(n|1)-action where aff(n|1) is the affine subalgebra of K(n). This result allows us to determine the superalgebra of endomrphisms on D n,k λ,μ commuting with the osp(n|2)-action for n ∈ {1, 2, 3} where osp(n|2) is the orthosymlectic superalgebras of S 1|n .