I'd like to ask if the following "analogue" of the fact $\mathbb{C}[\mathfrak{g}]^G$ for semisimple $G$ is a polynomial ring is known, or disproved, or anything.

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$. Let $\mathfrak{g}[[t]]=g\otimes \mathbb{C}[[t]]$, similarly for $\mathfrak{g}((t))$. Pick a nilpotent orbit $O$ of $\mathfrak{g}$, and consider a subset of $\mathfrak{g}((t))$ of the form

$X_O= \{n/t + x \ |\ n\in \overline{O}, \, x\in \mathfrak{g}[[t]]\} $.`

$\mathfrak{g}[[t]]$ acts naturally on $X_O$. I'd like to know if $R=\mathbb{C}[X_O]^{\mathfrak{g}[[t]]}$ is a polynomial ring. More precisely:

Functions on $X_O$ are generated by coordinate functions $a_i:X_O\to \mathbb{C}$ which is defined for $a\in \mathfrak{g}^*$ and $i\in\mathbb{Z}$ via $a_i(y)=$ ($a$ applied on the coefficient of $t^i$ of $y\in X_O$). We assign bidegree $(1,i)$ to $a_i$.

Then, I'm interested in the subring $R_-$ of $R$ whose second degree is negative. I think $R_-$ is a polynomial ring generated by $(\dim O)/2$ variables. I'd like to know the bidegree of the generators, too.

(One might need to require $\mathfrak{g}$ to be simply-laced, and/or $O$ to be special.)

For example, when $O$ is the orbit of zero, $R_-$ is just $\mathbb{C}$.

When $O$ is the regular nilpotent orbit, $R_-$ has $\dim O/2$ generators $c_{k,i}$ where $k=1,\ldots,\mathrm{rank}\, \mathfrak{g}$ and $i=1,\ldots, e_k$ where $e_k$ is the $k$-th exponent of $\mathfrak{g}$. Explicitly, $c_{k,i}$ is given as follows: denote by $c_k\in \mathbb{C}[\mathfrak{g}^*]^G$ a generator of degree $e_k+1$. Then we can consider $c_k(y)\in \mathbb{C}((t))$. for $y\in \mathfrak{g}((t))$. Finally we can define $c_{k,i}:X_O\to \mathbb{C}$ as the coefficient of $t^{-i}$ of $c_k(y)$ where $y\in X_O$.