Suppose that:

 - $X$ is a smooth complex algebraic variety,
 - $f : X \to D$ is a map to a small disc, smooth away from 0,
 - $Z_\epsilon = f^{-1}(\epsilon)$, and $Z = Z_0$.

Then there is a procedure (“nearby cycles”) which produces a complex $\psi_f(\mathbb{Q}_X)$ on $Z$ whose cohomology agrees with that of $Z_\epsilon$ for $\epsilon \ne 0$. (My notation is that $\mathbb{Q}_X$ is always shifted so as to be perverse.)
This belongs to a powerful toolbox of techniques to relate the cohomology of general and special fibres, with important applications in algebraic geometry, number theory and representation theory.

**Question:** Suppose that I just give you $Z \subset X$ but not $f$, can I “guess” $\psi_f(\mathbb{Q}_X)$?

Here is a rough proposal for how to do so, and I’m wondering if this is discussed somewhere in the literature. It seems a little like magic, and I might be making mistakes.

Firstly, $\psi_f(\mathbb{Q}_X)$ is a perverse sheaf, and it comes with a monodromy endomorphism $\mu$. **I assume that $\mu$ is unipotent**. Hence $N = 1-\mu$ is a nilpotent endomorphism of the nearby cycles, and I have a short exact sequence of perverse sheaves:

$$ 0 \to \phi_f(\mathbb{Q}_X) \stackrel{N}{\to}  \psi_f(\mathbb{Q}_X) \to i^*\mathbb{Q}_X \to 0$$

Thus, $i^*\mathbb{Q}_X$ is the “coinvariants of the monodromy”.

Secondly, $\psi_f(\mathbb{Q}_X)$ carries a weight filtration $W$, and a deep theorem of Gabber states that the weight filtration agrees with the monodromy filtration. In particular:
$$N^i : gr_W^{-i}(\psi_f(\mathbb{Q}_X)) \stackrel{\sim}{\to} gr_W^{i}(\psi_f(\mathbb{Q}_X))$$
is an isomorphism.

Now, assume that I know the successive subquotients of the weight filtration on $i^*\mathbb{Q}_X$. Then, it seems that the above gives a very good picture of the associated graded of the weight filtration on $\psi_f(\mathbb{Q}_X)$. Namely, every $IC_\lambda$ which occurs in $gr_W^{-i}(i^*\mathbb{Q}_X$) contributes an $IC_\lambda$ in weight filtration steps $-i, -i+2, \dots, i-2, i$ to the weight filtration on $\psi_f(\mathbb{Q}_X)$.

An analogy: any finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ is recoverable from its lowest weight vectors. Under this analogy, the lowest weight vectors are given by $i^*\mathbb{Q}_X$.

Some precise questions:
1) Under the above assumptions is it correct that I can recover the associated graded of the weight filtration on $\psi_f(\mathbb{Q}_X)$ from that of $i^*\mathbb{Q}_X$?
2) This appears to imply that $\psi_f(\mathbb{Q}_X)$ is automatically constructible for any stratification that makes $i^*\mathbb{Q}_X$ constructible, which surprises me a little.
3) This appears rather powerful. Has this technique been applied usefully somewhere? In other words, where can I read more?!