G_0(\log\,D) & := & \frac{H^0(V,\Omega^{n-1}_{V/T}(\log \,D)[\theta])}{(\theta d-df\wedge)H^0(V,\Omega^{n-2}_{V/T}(\log \,D)[\theta])}
&;&
G_0(*D) & := & \frac{H^0(V,\Omega^{n-1}_{V/T}(*D)[\theta])}{(\theta d-df\wedge)H^0(V,\Omega^{n-2}_{V/T}(*D)[\theta])}.
\\ \\
\end{array}$$ Then $G(*D)$ is ${{\mathds C}}[\theta,\theta^{-1},t,t^{-1}]$-free of rank $n$ and $G(\log\,D)$ (resp. $G_0(*D), G_0(\log\, D)$) is a ${{\mathds C}}[\theta,\theta^{-1},t]$- (resp. ${{\mathds C}}[\theta,t,t^{-1}]$-, ${{\mathds C}}[\theta,t]$-) lattice inside $G(*D)$. These modules fit into the following diagram $$\begin{array}{ccc}
G(\log\,D) & \subset & G(*D) \\
\cup & &\cup \\
G_0(\log\, D) &\subset & G_0(*D).
\end{array}$$ Define a connection $$\nabla:G_0(\log\,D)\longrightarrow G_0(\log\,D)\otimes \theta^{-1}\Omega^1_{{{\mathds C}}\times T}\left(\log(\{0\}\times T)\cup({{\mathds C}}\times \{0\})\right)$$ by putting, for a form $\omega\in H^0(V,\Omega^{n-1}_{V/T}(\log\,D))$, $$\label{eq:ConnectionOperator}
\begin{array}{rcl}
\nabla_{\partial_\theta}([\omega]) & := & \theta^{-2}[f\cdot\omega] \\ \\
\nabla_{\partial_t}([\omega]) & := & \frac1{nt}([\operatorname{\textup{Lie}}_E(\omega)]-[\theta^{-1} f\cdot\omega])
\end{array}$$ and extending by the Leibniz-rule (for $\nabla_{\partial_\theta}$) resp. $\theta$-linearly (for $\nabla_{\partial_t}$). We denote by $\nabla$ the induced connection on $G(\log\,D)$, $G_0(*D)$ and $G(*D)$.

One of the main results of [@dGMS] concerns the construction of various bases of the module $G_0(\log\,D)$ (hence, of all the other modules given above), such that the connection takes a particularly simple form. This can be summarized as follows.

\[prop:recalldGMS\] There is a ${{\mathds C}}[\theta,t]$-basis $\underline{\omega}^{(1)}=(\omega_1,\ldots,\omega_n)$ of $G_0(\log\,D)$ such that $$\label{eq:Basis}
\nabla(\underline{\omega}^{(1)}) = \underline{\omega}^{(1)} \cdot
\left[
(A_0\frac{1}{\theta}+A_\infty)\frac{d\theta}{\theta}+(-A_0\frac{1}{\theta}+A'_\infty)\frac{dt}{nt}
\right]$$ where $$A_0:=
  \begin{pmatrix}
    0 & 0&\ldots &0& c\cdot t\\
    -1   &0&\ldots&0& 0\\
    \vdots & \vdots & 