TITLE: D-modules on affine space that are regular at infinity
QUESTION [6 upvotes]: 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 $\infty$?

REPLY [3 votes]: 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.