from ${\sf St}_{\bm{k}}^C$ to itself.

For two distinct points $c\neq c'$ in ${\mathbb{C}}$, the Stokes direction of the pair is the set of points $\theta$ in $S^1$ such that $\Re e^{-i\theta}(c-c')=0$. For a finite set $C\subset {\mathbb{C}}$, an open interval $I\subset {\mathbb{R}}$ is called $C$-good if the image in ${\mathbb{R}}/{2\pi{\mathbb{Z}}}$ of $I$ contains exactly one Stokes direction for each pair $c\neq c'$ in $C$. We put $I_{\theta_\circ}:=]\theta_\circ-\frac{\pi}{2}-\varepsilon,\theta_\circ+\frac{\pi}{2}+\varepsilon[$ for fixed $\theta_\circ\in{\mathbb{R}}$ where $\varepsilon$ is a sufficiently small positive number . If $I_{\theta_\circ}$ is $C$-good for sufficiently small $\varepsilon >0$, $\theta_\circ$ is called $C$-generic. It is equivalent to the condition that $\theta_\circ+\pi/2$ is not the Stokes direction of any pair $c\neq c'$ in $C$.

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1.  On any $C$-good open interval $I\subset {\mathbb{R}}$, there exists a unique splitting $\eta_{[I]}:\L\mid_{[I]}{\xrightarrow{\sim}}{\mathrm{gr}}\L\mid_{[I]}$ compatible with the Stokes filtrations where $[I]$ is the image of $I$ in ${\mathbb{R}}/2\pi{\mathbb{Z}}$.

2.  Let $\lambda:(\L,\L_\bullet)\to(\L',\L'_\bullet)$ be a morphism in ${\sf St}_{\bm k}^C$. Then, for any $C$-good open interval $I\subset {\mathbb{R}}$, the restriction $\lambda|_{[I]}$ is graded with respect to the splittings in $(1)$. That is, the induced morphism $\lambda_{c, c'}:{\mathrm{gr}}_c\L\to {\mathrm{gr}}_{c'}\L$ is zero for any pair $c\neq c'$ in $C$.

Let ${{\sf Rep}}_{\mathbb{Z}}({\bm k})$ be the category of finite dimensional representations of 