Riemann Hilbert Correspondence with fixed stractification

Question

Riemann Hilbert Correspondence states for complex manifold $X$, the bounded derived category of $D$ modules on $X$ with cohomology being regular holonomic is equivalent to the bounded derived category of sheaves on $X$ with constructible cohomology.

My question is if we fix a particular stractification $\mathcal{S}$ for $X$ and require the cohomology sheaves is constructible with $\mathcal{S}$, what would be the corresponding subcategory on the $D$ modules side?

Answer 1

One should phrase the constructibility in terms of six-functors and then, since the RH correspondence respects those, one will see what is the corresponding notion.

First, to be a local system for a constructible sheaf translates under RH to the D-module being smooth (i.e., free of finite rank as an $\mathcal{O}$-module).

So, if constructibility of a sheaf $F$ means that for any stratum $i_{\alpha} : S_{\alpha} \to X$ one has that $i_{\alpha}^* F$ is a local system, the corresponding condition for a $D$-module $M$ will be that $i_{\alpha}^* M$ is smooth for all $\alpha$.

(Slightly orthogonally to the question itself; One also needs to be careful that there is a second notion - a sheaf $F$ is $!$-constructible if $i_{\alpha}^! F$ is a local system for all $\alpha$. In some situations (nice equivariant ones usually), $!$-constructibility is equivalent to $*$-constructibility.)