was carried out for $G={\mathrm{SL}}_2({\mathbb{R}})$ in [@cmz] (see also [@ga]).

The following theorems are proven in [@do].

For all $\d \not=1,$ there exists an isomorphism $${\cal Q}_{\l,\mu}: {\cal S}_{\d}^1(M)\oplus {\cal S}_{\d}^0(M)  
\rightarrow {{\mathcal{D}}}_{\l,\mu}^1(M),$$ given as follows: for all $P=P^i\xi_i+P_0\in~{\cal S}_{\d}^1(M)
\oplus {\cal S}_{\d}^0(M),$ one can associate a linear differential operator given by $$\begin{aligned}
\label{yah1}
{\cal Q}_{\l,\mu} (P)&=&
P^{i}\n_i +\a \n_i P^{i}+P_0,\end{aligned}$$ where $$\alpha=\displaystyle \frac{\l}{1-\delta}\cdot$$ This map does not depend on the rescaling of the metric, intertwines the action of ${\mathrm{Diff}}(M).$

For $n>2$ and for all $\d \not=\frac{2}{n}, 
\frac{n+2}{2n}, \frac{n+1}{n},\frac{n+2}{n},$ there exists an isomorphism $${\cal Q}_{\l,\mu}: {\cal S}_{\d}^2(M) \rightarrow {{\mathcal{D}}}_{\l,\mu}^2(M),$$ given as follows: for all $P=P^{ij}\xi_i\xi_j \in~{\cal S}_{\d}^2(M),$ one can associate a linear differential operator given by $$\begin{aligned}
{\cal Q}_{\l,\mu} (P)&=&
P^{ij}\n_i\n_j \nonumber\\[2mm]
&&+(\a_1 \n_i P^{ij}+\a_2\,\g^{ij}\, \g_{kl}\n_i P^{kl})
\n_j 
\label{Tensor1}\\[2mm]
&&+\a_3\n_i\n_j P^{ij}+\a_4\, \g^{st}\, \g_{ij}\n_s\n_t P^{ij}
+\a_5 R_{ij}P^{ij}+\a_6 R \,\g_{ij}\,P^{ij}, 
\nonumber\end{aligned}$$ where $R_{ij}$ (resp. $R$) are the Ricci tensor components (resp. the scalar curvature) of the metric $\g,$ the constants $\a_1,\ldots,\a_6$ are given by $$\begin{array}{ll}
\a_1=\displaystyle 
\frac{2(n\l+1)}
{2+n(1-\d)}, &\a_2= \displaystyle 
\frac{n(\l+\mu-1)}
{(2+n(1-\delta ))
(2-n\d)},\\[3mm]
\a_3= \displaystyle 
\frac{n\l(n\lambda+ 1)}
{(1+n(1-\delta))(2+n(1-\delta ))}, & 
\a_5= \displaystyle
\frac{n^2\l(\mu-1)}
{(n-2)(1+n(1-\delta))} ,\\[3mm]
\a_4= \displaystyle 
\frac{n\l(n^2 \mu (2-\l-\mu)+2(n\l +1)^2-n(n+1))}
{(1+n(1-\delta))(2+n(1-\d))(2+n(1-2\d))(2-n\delta )}, & 
\a_6= \displaystyle
\frac{(n\d-2)}
{(n-1)(2+n(1-2\delta))}\, \a_5 \cdot\\[3mm]
 &\\
