Let $Delta$ be a small disk, and let $M$ be a regular holonomic right $mathcal{D}_Delta$-module. Let $z$ be the coordinate on $Delta$, and $kgeq 1$. Then we get the $V$-filtration on $M$ along $z^k$:

$$V^bullet_{z^k}(M).$$

Recall that since for $k>1$ the function $z^k$ is not smooth, this is defined as

$$V^alpha_{z^k}(M):= (Motimes 1)cap V^{alpha}_t(i_{k+}M),$$

where $i_k:Deltato Delta^2$ is the map $cmapsto (c,c^k).$

I would guess that:

$$V^{alpha}_{z^k}(M) = V^{kalpha}_z(M),$$

but I cannot prove it.

Question 1: is this correct?

Besides not being able to prove it, the result in this question suggests that it might not be true. This result shows that $V^alpha_{t}(i_{k+}M)$ is generated over $mathcal{D}_Delta$ by

$$M^nu otimes partial_t^j ~~textrm{with $nu leq k(alpha-j)$},$$

where

$$M^nu:={min Mmid u(zpartial_z-nu)^p=0text{ for }pgg 0}$$

Now working this out means that we expect that

$$V^{alpha}_{t}(i_{k+}M)subsetneq V^{kalpha}_{t}(i_{1+}M).$$

However, I am not sure that this also automatically means that

$$V^{alpha}_{z^k}(M)=V^{alpha}_{t}(i_{k+}M)cap (Motimes 1)subsetneq V^{kalpha}_{t}(i_{1+}M)cap (Motimes 1)=V^{kalpha}_z(M)?$$

More generally I am not sure that

$$gr_{V_t}^alpha(i_+M)not=0 Leftrightarrow gr_{V_f}^alpha(M)not=0.$$

Where $i$ again denotes the graph embedding along $f$.

Question 2: is this equivalence true?

Notice that this does not follow from Kashiwara’s equivalence, since the filtered pieces in the $V$-filtration are not $D_X$ and $D_{Xtimes mathbb{C}}$-modules.