Regular Functions on Slodowy Slices

Question

I recently learned that there is a natural identification, given a simple Lie algebra,

Polynomial functions on the Slodowy slice of a regular nilpotent orbit $\simeq$ polynomial functions on the Cartan subalgebra invariant under the Weyl group.

I guess there's a generalization of this statement for subregular or other nilpotent orbits. Which book/article should I have a look at? For example, I guess there's a equivalence of the form

Polynomial functions on the Slodowy slice of a subregular nilpotent orbit $\simeq$ polynomial functions on the Levi subalgebra of the form $sl_2\times \mathbb{C}^{r-1}$ satisfying conditions *

What are the conditions * I need to fill in the statement above?

Answer 1

The Slodowy slice to a nilpotent $e$ is by definition the set of elements $x$ such that $x-e$ commutes with some fixed $f$ such that $e$ and $f$ generate an $\mathfrak{sl}_2$. Thus, it is an affine space modeled on the commutant of $f$, a Lie subalgebra of $\mathfrak{g}$.

In the regular case, this subalgebra isn't a torus (it's nilpotent), though it is a flat limit of tori (just as a regular nilpotent is a limit of semi-simples). In $\mathfrak{sl}_n$, if we take the principal nilpotent, its commutant is the space of matrices with 0's on or below the diagonal, and with entries constant on every shifted diagonal.