There is a close but not perfect relationship between algebraic D-modules on C^n, constructible sheaves on C^n in the analytic topology, and \ell-adic sheaves on an n-dimensional vector space over a field of characteristic p:
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Forming the sheaf of not-necessarily-algebraic solutions to the algebraic system of differential equations carries a class of D-modules (regular holonomic D-modules) to constructible sheaves, and in an appropriately derived setting this is an equivalence of categories. This is the Riemann-Hilbert correspondence.
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Less functorially, you can try to find an \ell-adic model for your constructible sheaf on affine space over not C but something like the p-adic integers (which you embed in C), and reduce mod p.
Many things can go wrong, but there are comparison theorems. For instance there is a good notion of "constant sheaf on a subvariety" in all three settings, and taking the cohomology of this sheaf gives a similar answer in the three cases: this is the comparison theorem between de Rham, Betti, and \ell-adic cohomology.
In all three settings there is an operation called Fourier transform. An algebraic D-module on C^n is a module over the Weyl algebra C[x,y,…,d/dx,d/dy,…], and its Fourier transform is the pullback along the change of variables x –> -d/dx, d/dx –> x. In the topological setting you have the Fourier-Sato/Kashiwara-Schapira transform, whose target is sheaves on the real dual vector space to C^n. And in characteristic p you have the Fourier-Deligne transform, which involves the Artin map x^p – x somehow.
For both D-modules and \ell-adic sheaves, there is no restriction on the kind of sheaf you can transform. But it may take a D-module with regular singularities to one without regular singularities, or take an \ell-adic sheaf without wild ramification to one with wild ramification. In the topological setting, you can only take the Fourier transform of sheaves that are constant along rays from the origin (usually because these sheaves are C^*-equivariant), but then the new thing is as well-behaved as the old.
I would like to understand better how these things are related, or what can go wrong. Maybe they just have misleading names–I am pretty sure that the so-called Fourier-Mukai transform is a red herring, here. But I have seen them used in the same way in Springer theory: to construct representations of Weyl groups on the cohomology (Betti or \ell-adic) of Springer fibers. Are there any comparison results between the different Fourier transforms?
PS Ben's suggestion is that there should be very strong comparison theorems if we work with C^* or G_m-equivariant objects in all settings. So, specifically, the Riemann-Hilbert correspondence should commute with the Fourier transform on C^*-equivariant holonomic D-modules and constructible sheaves. Is this a well-known and referencable result?
Best Answer
I strongly believe they are the same (i.e. commute with Riemann-Hilbert) for $\mathbb{G}_m$ equivariant things but have no reference or proof for this fact.
My (unprecise) argument is that both of them involve replacing a sheaf on a vector space V by the sheaf on the dual space $V^*$ given by taking a covector, and doing vanishing cycles at the origin with respect to it.
They way you should think about this for $\ell$-adic Fourier transform is that the stalks of the $\ell$-adic transform at covectors are the hypercohomology of the original sheaf tensored with the Artin-Schreier sheaf for that covector. Since the Artin Scheier sheaf snuffs out anything which is constant along the different values of the covector, it only picks up bizarre jumps, which by $\mathbb{G}_m$-equivariance can only happen at the origin. What else measures bizare jumps along the values of a covector? Vanishing cycles!