Symmetries of modules of differential operators on the supercircle $$S^{1|n} $$

Abstract

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 .