# D-modules on affine space that are regular at infinity

If I have a D-module $M$ on $\mathbb{A}^n$ (which is essentially the same thing as a module over the Weyl algebra), then I can push this D-module forward to $\mathbb{P}^n$ to get a D-module $j_*M$ on the projective closure.

Is there an intrinsic description of those $M$ such that $j_*M$ are regular on the divisor at $\infty$?

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I think Mebkhout has something that does this. – Avi Steiner Feb 15 at 6:01
@AviSteiner That's nice. Unfortunately, I've now completely forgotten what inspired me to ask this question... – Ben Webster Feb 15 at 16:55

In dimension one, $j_*M$ is regular at infinity iff the Fourier transform $F(M)$ has no singularity outside 0 and $\infty$. The reason is obviously that Fourier exchanges the D-modules $\delta_c = D/D(t-c)$ et $Oe^{ct} = D/D(\partial_t-c)$. There's probably a generalization for higher dimensions but I've never seen it.
This is a great answer, but I am confused by the reason is obviously' sentence... is this supposed to prove the if and only if' claim? If so I would like to see more explanation (to me it seems that some facts about local Fourier transform would be required). – t3suji May 20 '11 at 2:21
Come to think of it, is this even true? Consider the D-module with one generator f and one relation $\partial f=t f$. (One can write it as $Oe^{t^2/2}$ using YBL's notation.) This $D$-module has an irregular singularity at infinity, but its Fourier transform is essentially itself, so it has no singularities outside $0$ and $\infty$. – t3suji May 20 '11 at 15:17