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One thing you could say is that a homomorphism $\pi_1(M) \to SL(n,\mathbb{C})$ is the same as a flat connection on a principal $SL(n,\mathbb{C})$-bundle over $M$. Flat connections are important in physics?
One thing you can say is that D-modules finitely generated over the structure sheaf are the same as flat vector bundles which are the same as representations of the fundamental group.
I am aware of Motsak's thesis. (s)he actually gives explicit generators for $Z(U(\mathfrak{gl}_n))$ but they do not have the correct character. You could go from those generators to the ones I am looking for using symmetric function theory, but I don't think that would be easy
Have you thought about what happens geometrically? Suppose that you have an algebraic variety $X$ and a vector bundle $ \pi : E \to X$. If $s$ is a section that $Z = \{ s(x) = 0 \} $ defines a subset of $X$. If $E$ is trivial over $U$, then $Z \cap U$ is cut out by ${\rm rank} \, E$ equations. It follows that the whole set $Z$ is closed in the Zariski topology.
One situation where you need to use the elements directly is when you map into a tensor product. This happens a lot when you think about Morita theory. Often the indecomposable projectives that you are working with are induced up from some smaller ring(oid) and you need to compute the maps between these indecomposable projectives
Let $f : A \to B$ be a map of rings and $M$ an $A$-module. Then $B \otimes_A M$ is a pointwise left Kan extension computed as a coend. The universal property of tensor products is useful, but often it is useful to manipulate elements in $B \otimes_A M$ directly. More generally, if you repace $B$ and $A$ with ringoids (something that I do frequently) it is still useful to do explicit computations instead of just relying on the universal properties.
