Cohomology of $\mathfrak {osp}(1|2)$ acting on linear differential operators on the supercircle $S^{1|1}

We compute the first cohomology spaces $H^1(\mathfrak{osp}(1|2);\mathfrak{D}_{λ,μ})$ ($λ, μ\in\mathbb{R}$) of the Lie superalgebra $\mathfrak{osp}(1|2)$ with coefficients in the superspace $\mathfrak{D}_{λ,μ}$ of linear differential operators acting on weighted densities on the supercircle $S^{1|1}$. The structure of these spaces was conjectured in \cite{gmo}. In fact, we prove here that the situation is a little bit more complicated. (To appear in LMP.)

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