A quick observation: You can use Szemerédi's theorem to reduce to the case where (for some fixed large n) $a_0,\cdots, a_n$ is a permutation of $0,\cdots n$. I don't immediately see how to do this case though.

...The mention of "geometric Langlands" is a red herring. My suggestion boils down to the observation that, in some ways, arithmetic galois groups and (complex) differential galois groups behave similarly. This has been known at least since 1970. (I don't think Langlands had this in mind for his letter to Weil.) But I would be kind of shocked if Langlands was thinking of something else for the link he mentions to the geometric Langlands program. The moment you start thinking about ramification there you instantly run into the theory of irregular singularities.

@WaleedQaisar Well, I am not sure what Langlands is referring to in his letter to Weil. Langlands only says that there are "traces of this topic", and I can imagine various ways to see traces (e.g. paul garrett's comments seem plausible to me) but it is certainly not front and center in his letter. That being said, I would say that the link I previously mentioned is very concrete...

It sounds like you care only about algebraic representations. In that case, you don't have to worry about anything in 4 or 5 - those are for non-algebraic representations. So if you are looking at representations of $G(\mathbb{R})$, this is happening in a different category (Lie group representations) and so you get many more (infinite-dimensional) representations. Unitarity is another concept that isn't very applicable in the algebraic setting.

My guess is the comment has rather concrete meaning. The study of irregular singular points of ODE is more or less the study of $\mathcal{D}$-modules on the formal disk. The space of local Langlands parameters in geometric Langlands is a moduli space of $\mathcal{D}$-modules on the formal disk. So the study of singularities of ODE is the geometric counterpart of the study of ramification in the arithmetic setting.

When $\beta$ is an actual homomorphism, this construction is equivalent to taking ordinary Lie algebra homology of a twist of $M$. I am unaware of any way to define your functor when $\beta$ is an arbitrary element of $\mathfrak{g}^*$. BTW, perhaps the most prominent example of this is in the case where your Lie algebra is a maximal nilpotent $\mathfrak{n}$ inside a semisimple Lie algebra, e.g. (finite) Drinfeld-Sokolov reduction. But there the interest comes from the fact that you assume that your $\mathfrak{n}$-modules have more structure (usually you consider $\mathfrak{g}$-modules).

(I suspect that your rational/irrational bifurcation is motivated by the case of affine Kac-Moody algebras, where rationality of the level controls what category O looks like. There it appears because you have the entire affine Weyl group to play with, but here you only have the finite Weyl group and so the possibilities are more limited.)

IIRC the story is very similar to the integral case, except that instead of $W$ you need to consider the subgroup $W'$ generated by those simple reflections $s_{\alpha}$ that send $\lambda$ to $\lambda+n\alpha$ for some integer $n$. Then if $\lambda$ is $W'$-dominant then Beilinson-Bernstein holds and there is a theory of reflection functors that takes care of the other weights. But I might be misremembering/confusing with something else.

@HarryGindi Can you point to a specific lemma/theorem? I am seeing many uses of the derived category QCoh, but I cannot find any part where he is using the abelian category for a general prestack.

@HarryGindi I'm not seeing where in SAG he uses the abelian category QCoh for a general prestack X (as opposed to in specific cases e.g. X is an actual geometric stack.) Can you give me the number?

@lush By avoiding the use of Zariski covers, you are more or less trying to prove this statement for presheaves $X$ rather than schemes $X$. As you noted, the statement still holds true in this setting by generalities on presentable categories. However, the category in question is badly behaved for general presheaves - to construct a good category you need to work in a dg setting - and to the best of my knowledge is never used in this generality. So I recommend not avoiding Zariski covers.

I'm afraid that this result is not true (it fails for e.g. a nodal curve. One way to see this is to compute the singular cohomology over $\mathbb{C}$.) The issue is that the math.SE link you give is only for the Zariski topology, not the etale topology (see the comment of Roland.)

@DanielLitt good point!

@jg1896 Sorry, by counter-example here I actually mean example where $X$ is birational to $X/G$. The reason is that in the general type case for most $X$ and $G$ it will be possible to distinguish $X$ and $X/G$ via natural invariants, e.g., the $h^{0,i}$.

Q2 maybe becomes more interesting if you ask for X general type, rather than X non-rational... it's not immediately clear to me whether there should be an easy counterexample.