Symmetries of modules of differential operators

Let ${\cal F}\_λ(S^1)$ be the space of tensor densities of degree (or weight) $λ$ on the circle $S^1$. The space ${\cal D}^k\_{λ,μ}(S^1)$ of $k$-th order linear differential operators from ${\cal F}\_λ(S^1)$ to ${\cal F}\_μ(S^1)$ is a natural module over $\mathrm{Diff}(S^1)$, the diffeomorphism group of $S^1$. We determine the algebra of symmetries of the modules ${\cal D}^k\_{λ,μ}(S^1)$, i.e., the linear maps on ${\cal D}^k\_{λ,μ}(S^1)$ commuting with the $\mathrm{Diff}(S^1)$-action. We also solve the same problem in the case of straight line $\mathbb{R}$ (instead of $S^1$) and compare the results in the compact and non-compact cases.

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