Symmetries of modules of differential operators

Abstract

Let ${\cal F}\_\lambda(S^1)$ be the space of tensor densities of degree (or weight) $\lambda$ on the circle $S^1$. The space ${\cal D}^k\_{\lambda,\mu}(S^1)$ of $k$-th order linear differential operators from ${\cal F}\_\lambda(S^1)$ to ${\cal F}\_\mu(S^1)$ is a natural module over $\mathrm{Diff}(S^1)$, the diffeomorphism group of $S^1$. We determine the algebra of symmetries of the modules ${\cal D}^k\_{\lambda,\mu}(S^1)$, i.e., the linear maps on ${\cal D}^k\_{\lambda,\mu}(S^1)$ 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.