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?