Unfortunately this question will be a bit vague, since the question revolves around a memory of something I may have heard in a talk (long time ago).

Let $X$ be a smooth complex variety. Let $MHM(X)$ be Saito's category of mixed Hodge modules on $X$. Let $M\in MHM(X)$. Part of the data of $M$ consists of $D$-module with a canonical good filtration (the Hodge filtration). In particular, on taking the associated graded of this filtration I can obtain a quasi-coherent sheaf on $T^*X$ (even $\mathbb{C}^*$-equivariant but for the moment let me ignore that).

What sort of full/faithfulness properties does taking this associated graded have? Of course, in general it's not going to be nicely behaved, but can something nice be said if I am only looking at morphisms/extensions between irreducible mixed Hodge modules (basically irreducible perverse sheaves).

I seem to vaguely recall a talk in which the speaker mentioned something along these lines and attributed it to Simpson. Does this ring any bells with anyone?

Note there are a number of different categories and extension groups floating around here. For instance, there are extension groups in $D(MHM(X))$, $D(X)$, $D_{Qcoh}(T^*X)$ and $D_{Qcoh}^{\mathbb{C}^*}(T^*X)$. Worse there's two different $D(X)$'s here. One is with $\mathbb{C}$ coefficients and the other with rational coefficients (for a Hodge module you need an underlying rational or real structure).


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