Let $G:=\operatorname{GL}_{N}$ act on its Lie algebra $\mathfrak{g}:=\mathfrak{gl}_{N}$ by conjugation. Then it acts naturally on the associated ring $\mathcal{O}(\mathfrak{g})$ of (algebraic or analytic) functions on $\mathfrak{g}$, yielding the associated GIT quotient $\mathcal{O}(\mathfrak{g})//G$, namely the spectrum $\operatorname{Spec}(\mathcal{O}(\mathfrak{g})^{G})$ of the $G$-invariant functions. Since the regular semisimple elements of $\mathfrak{g}$ are dense in $\mathfrak{g}$, this ring is the (algebraic or analytic) symmetric functions in the eigenvalues of the variables from $\mathfrak{g}$, so it is essentially the symmetric power $\operatorname{Sym}^{N}\mathbb{A}^{1}$.
I'm interested in the tangent space of this variety, in the natural variables (namely functions of the eigenvalues). On the regular semisimple locus, the map from $\mathbb{A}^{N}$ to that symmetric power is a local isomorphism, so that the usual tangent space to $\mathbb{A}^{N}$ at such points does the trick. However, on the non-regular locus, this is no longer the case, and it seems that the natural elements from $\mathfrak{g}$ no longer produce the full tangent space. In fact, one needs some "fractional powers of the differentials" for getting the full tangent space. This is, at least, what my investigations of this question produced.
My question is - is there any reference dealing with this object? I'd like to compare what I got to the existing literature (if there is such), and see whether there are better ways to view these tangent spaces than what I received. Thanks!
Edit: Following the answer by @Sasha, I'm making things more precise. If the point on $\operatorname{Sym}^{N}\mathbb{A}^{1}$ arises from taking $a_{i}\in\mathbb{A}^{1}$ with multiplicities $m_{i}$ summing to $N$, then the abstract argument presented in this answer gives, as expected, $m_{i}$ coordinates of that space that are "centered at $a_{i}$".
Recalling that for varieties like $\mathbb{A}^{N}$, elements of the tangent space at a (closed) point $a$ act like differential operators on functions $\mathbb{A}^{N}$ and evaluation at $a$, I'm interested in having the elements from above that are "centered at $a_{i}$" as explicit differential operators on functions on $\operatorname{Sym}^{N}\mathbb{A}^{1}$, viewed as symmetric functions on $\mathbb{A}^{N}$, and evaluating said derivatives at $a_{i}$.
Explicitly, at a point in $\mathbb{A}^{N}$ where all the coordinates differ, the associated subscheme $Z$ of $\mathbb{A}^{1}$ has $N$ distinct points, and the space of sections from @Sasha's answer consists of a 1-dimensional space over each point. If we lift a function on $\operatorname{Sym}^{N}\mathbb{A}^{1}$ to a symmetric function on $\mathbb{A}^{N}$, and substitute the $i$th point of $Z$ (in some order) as the $i$th variable, then a generator of the $i$th 1-dimensional space is the operator differentiating the resulting (symmetric) function with respect to the $i$th variable.
When points coincide though, such derivatives don't span the tangent space anymore, and higher order derivatives are expected. In @Sasha's description, this corresponds to the fact that over some points in $Z$, the length of $O_{Z}$ is larger. So my question is what are the differential operators on symmetric functions on $\mathbb{A}^{N}$ that correspond to the other sections of $O_{Z}$?
