This post is rather wordy and speculative, but I promise there is a concrete question embedded within. For experts, I'll open with a question:
Question: Given a compact Riemann surface $X$, why does one prefer the category of D-modules on the space/stack $Bun_{X}(G)$ instead of the (Fukaya) category of Langrangians on the space/stack of $G$-local systems $\mathcal{L}_{X}(G).$
Much of what follows is explaining what I mean by this question.
Let $G$ be a connected reductive complex Lie group and $G^{\vee}$ the Langlands dual group. Let $X$ be a compact Riemann surface (smooth projective curve over $\mathbb{C}$). The central players in the geometric Langlands conjecture are the stacks of $G$-bundles on $X,$ denoted $Bun_{X}(G),$ and the stack of $G^{\vee}$-local systems on $X,$ which I'll denote by $\mathcal{L}_{X}(G^{\vee}).$
The best hope (using the words of Drinfeld) is that there is a (derived) equivalence between the category of D-modules on $Bun_{X}(G)$ and the category of quasi-coherent sheaves on $\mathcal{L}_{X}(G^{\vee}).$ This best hope is by now entirely dashed, and, to name one main player, Gaitsgory has expended a tremendous amount of effort to both indicate its failure, and conjecture a solution, along the way writing some very technical and interesting papers to give evidence.
Meanwhile, Witten, Kapustin, and others have made efforts to indicate how to restore some symmetry to this nebulous blob of conjectures, by arguing that they have natural interpretations as dimensional reductions of certain four dimensional super-symmetric gauge theories.
In the course of reading about various aspects of this story, I was struck by a point that Witten has made many times in talks and in print that I want to ask about here. I'm going to speak rather prosaically from here on forward, and welcome an explanation about why my simplifications don't make any sense.
Starting from Hitchin's work in the late 1980's, it became clear that the space of $G$-local systems $\mathcal{L}_{X}(G)$ is basically the cotangent bundle of $Bun_{X}(G).$ If you take stacky language seriously enough, this can be made reasonably precise. Taking this as true, there are many concrete, though not immediately applicable, theorems which indicate that the category of D-modules on $Bun_{X}(G)$ is equivalent to the Fukaya category of Langrangians in its co-tangent bundle: the latter of which is $\mathcal{L}_{X}(G).$
Using this quasi-logic, the best hope geometric Langlands conjecture posits an equivalence of (derived) categories between the Fukaya category of $\mathcal{L}_{X}(G)$ and the category of quasi-coherent sheaves on $\mathcal{L}_{X}(G^{\vee}).$
In other words, geometric Langlands becomes the statement that $\mathcal{L}_{X}(G)$ and $\mathcal{L}_{X}(G^{\vee})$ are mirror partners in the sense of homological mirror symmetry.
Question: Has this interpretation been taken seriously somewhere in the mathematical literature, and if not, is there a good reason it hasn't been?
As one final comment. There is (at least) one major advantage to putting the conjecture into this language. Both the Fukaya category of $\mathcal{L}_{X}(G)$ and the category of quasi-coherent sheaves on $\mathcal{L}_{X}(G^{\vee})$ can be defined without recourse to the complex/algebraic structure on $X.$ This is because both of these spaces/stacks are naturally complex symplectic, and this structure is independent of the complex/algebraic structure on $X.$
With this in mind, it makes more sense to refer to $\mathcal{L}_{\Sigma}(G)$ and $\mathcal{L}_{\Sigma}(G^{\vee})$ where $\Sigma$ is a connected, oriented, smooth surface. An advantage of this is that now topological symmetries (diffeomorphisms) of $\Sigma$ act naturally on these spaces/stacks and the subsequent categories. The study of these symmetries is what people in my field call the study of the mapping class group of $\Sigma,$ and there are many deep open questions about the mapping class group, that might find a natural home in the aforementioned discussion.
Best Answer
To answer [a paraphrase of] your second question first: yes the Kapustin-Witten perspective on geometric Langlands has I think been taken very seriously by a segment of the math community. I find it very misleading though to say (as is often done) that "geometric Langlands is mirror symmetry for the Hitchin space" -- mirror symmetry is a statement about 2d TFTs, while geometric Langlands is one about 4d TFTs which implies a vast amount more structure -- specifically the most important structure for the Langlands story, the action of Hecke operators, is part of the 4d story but not of the mirror symmetry statement.
In any case as Will mentions the Betti Geometric Langlands conjecture formulated in https://arxiv.org/abs/1606.08523 is a direct response to your question - in particular to have a version of geometric Langlands which as you ask should depend only topologically on the Riemann surface (so eg have a mapping class group symmetry). However it is not formulated in Fukaya category language directly. I'm fairly ignorant of Fukaya categories but my impression is that the vast technical difficulties in the subject prevent them currently being rigorously defined on the kind of spaces we are talking about here -- namely both singular and stacky. So the Fukaya-theoretic conjecture you discuss is still more of a guiding principle than a precise question.
Also since the Hitchin space is noncompact you have to decide what KIND of Fukaya category you'd mean (assume say we are dealing with a smooth manifold), i.e. what conditions to put at infinity as Will says - infinitesimal, wrapped, or partially wrapped ("with stops"). The Betti conjecture, which Nadler and I felt captured the spirit of Kapustin-Witten, is morally taking the Fukaya category with stops in the direction of the Hitchin base - i.e. your prototypical Lagrangians allowed are Hitchin fibers, not sections. [By the way for one of probably quite a few papers I can't remember this instant which does treat aspects of GL in a Fukaya perspective there's Nadler's paper on Springer theory https://arxiv.org/abs/0806.4566]
So what to do instead of Fukaya categories? by the microlocal perspective of Nadler, Zaslow, Kontsevich,.... we expect to replace Fukaya categories with categories of microlocal sheaves, eg for cotangent bundles, with constructible sheaves on the base (and you can impose singular support conditions for the "stops" or growth conditions on the Lagrangians). This actually gets you very close to the original characteristic p origin of the geometric Langlands correspondence, which dealt with l-adic sheaves -- via the Grothendieck function-sheaf dictionary those are natural "categorified" substitutes for functions, eg automorphic functions. There are no D-modules in this story.
The beautiful D-module version developed by Beilinson-Drinfeld and Arinkin-Gaitsgory in particular -- the de Rham geometric Langlands correspondence -- has a quite different flavor in many respects, and I would claim is one step further from both the arithmetic (l-adic) origins and from the mirror symmetry story. It is motivated by two (closely related) stories -- the Beilinson-Bernstein realization of representations of Lie algebras as D-modules, which gives it a very close relation to representation theory of affine Kac-Moody algebras; and conformal field theory (eg theory of vertex algebras). This allowed Beilinson-Drinfeld to leverage a crucial result of Feigin-Frenkel to prove a "big chunk" (half-dimensional slice) of the de Rham conjecture, and Gaitsgory and collaborators to develop an amazing program to understand and solve the conjecture in general.
[As a side note I think it's overly dramatic to say the "best hope" is "dashed" -- rather the distance from the original dream to the Arinkin-Gatisgory formulation is technical and not very large and not terribly unexpected - though led to some beautiful math - and is purely about understanding how to match growth conditions on one side with singularity conditions on the other, just as in studying the Fourier transform in different function spaces.]
The de Rham story also has deep relations to physics. The physics I would say is somewhat insensitive to the de Rham vs Betti distinction, which is about different algebraic structures underlying an analytic equivalence, but many of the mathematical questions require you to pick your setting more precisely (except the "core" ones that live in the intersection of the two conjectures). The de Rham story comes up naturally in relation to CFT, to things like gauge theories of Class S and the AGT conjecture, and a whole world that is part of.
OK this is now way too long.