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To reuse your elliptic curves example: the moduli stack is a Deligne-Mumford stack, whose underlying moduli space is the modular curve; the stack remembers the extra data of the elliptic points and their finite automorphism group. If you look at Diamond-Shurman, they study the moduli curves in great detail without ever saying the word stack; this is fine since most of the book is focused on weight 2 modular forms, which happen to be the same as differentials on the modular curve. However, modular forms of higher weight should really be thought of as section of line bundles on the moduli stack.
It's hard to answer, because whether or not it's important to remember automorphisms depends very much in the context and what you want to do. In geometry, this distinction between groupoids and equivalence relations becomes the distinction between moduli stacks and coarse moduli spaces. If you want to think about families of objects, then you absolutely need to remember automorphisms and think about the moduli stack. However, the coarse moduli space is a much more concrete object to think about and is enough for some purposes, so sometimes you don't want to bother with stacks.
Isn't this limit just $E$ equipped with the zero Higgs field? in the case I think the answer is no as the underlying vector bundle of a stable Higgs bundle need not be stable.
The formula you mention in the update section is quite interesting, do you have a proof of it? It seems possibly somewhat related to the Chevalley-Solomon formula (mathoverflow.net/q/467091/160416), though I can't see off the top of my head a way to prove one from the other.
Note that $n=4$ is also qualitatively different from $n\ge 5$ since the star quiver with four leaves is an affine Dynkin diagram, so has tame representation type (while the representation type is wild for $n\ge 5$).
Also, am I understanding correctly that the nonabelian Hodge correspondence for $GL_n$ on the locus of stable bundles follows immediately from Narasimhan–Seshadri by the same reasoning, but that you need to do more work to understand the semistable case?
I think I see, thanks. I guess the fact that there exists a unique unitary connection on every line bundle follows from the Narasimhan–Seshadri theorem, but is there an easy way to see it for line bundles?
That's fair, I can believe that this isomorphism is something really special to $K=\mathbb{C}$. I'm still curious about the pro-unipotent quotients though.
I'm also a bit confused about where the $\mathbb{G}_m^2$ comes from. The moduli space of $1$-dimensional vector bundles with flat connection on an elliptic curve $X$ is an affine space bundle over $\operatorname{Jac}(X)=X$, not $\mathbb{G}_m^2$.
Can you say a bit more on how formality of the de Rham algebra yields an isomorphism of the pro-unipotent quotients? Regarding your second point, I'm not sure why that forbids an isomorphism of the pro-algebraic groups; in general the set of 1d representations of a pro-algebraic group doesn't come with a natural moduli space structure.
Why do you think that it is not the complex analytic space associated with the base change of $X$ to $\mathbb{C}$? That seems to me like the obvious interpretation.
