is given by the equation $$f_{\l,\m}(A)=f^*\circ A\circ {f^*}^{-1},
\label{Opaction}$$ where $f^*$ is the action (\[den\]) of ${\mathrm{Diff}}(M)$ on ${{\mathcal{F}}}_\l$.

By differentiating this action, one can obtain the action of the Lie algebra ${\mathrm{Vect}}(M).$

The formul[æ]{} (\[den\]) and (\[Opaction\]) do not depend on the choice of the system of coordinates.

Denote by ${\cal D}^2_{\l,\mu}(M)$ the space of second-order linear differential operators with the ${\mathrm{Diff}}(M)$-module structure given by (\[Opaction\]). The space ${\cal D}_{\l,\mu}^2(M)$ is in fact a ${\mathrm{Diff}}(M)$-submodule of ${{\mathcal{D}}}_{\l,\m}(M).$

[The space of Sturm-Liouville operators $\frac{d^2}{dx^2}+u(x): {{\mathcal{F}}}_{-1/2}\rightarrow {{\mathcal{F}}}_{3/2}$ on $S^1,$ where $u(x)\in 
{{\mathcal{F}}}_{2}$ is the potential, is a submodule of ${{\mathcal{D}}}_{-\frac{1}{2},\frac{3}{2}}^2(S^1)$ (see [@wi]). ]{}

The module of symbols
---------------------

The space of symbols, ${\mathrm{Pol}}(T^*M),$ is the space of functions on the cotangent bundle $T^*M$ that are polynomials on the fibers. This space is naturally isomorphic to the space ${\cal S}(M)$ of symmetric contravariant tensor fields on $M.$ In local coordinates $(x^i,\xi_i)$, one can write $P\in {\cal S}(M)$ in the form $$P=\sum_{l\geq 0}P^{i_1, \ldots,i_l}\xi_{i_1}\cdots \xi_{i_l},$$ with $P^{i_1, \ldots,i_l}(x)\in C^{\infty}(M).$\
We define a one parameter family of ${\mathrm{Diff}}(M)-$module on the space of symbols by $${\cal S}_{\d}(M):={\cal S}(M)\otimes {{\mathcal{F}}}_{\d}.$$ For $f\in {\mathrm{Diff}}(M)$ and $P\in {\cal S}_{\d}(M),$ in a local coordinate $(x^i)$, the action is defined 