There was a conjecture of Grothendieck, proved by Brieskorn in 1970, which says that:
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$ of ADE type, and $x\in \mathfrak{g}$ a subregular nilpotent element. Let $S\subset \mathfrak{g}$ be a transversal (in $x$) slice to the $G$-orbit of $x$. Then the restriction $(S,x)\to (\mathfrak{t}/W,0)$ of the adjoint quotient is a semi-universal deformation of the corresponding simple ADE singularity.
For other types the situation is a bit trickier, but is discussed in Slodowy, Four lectures on simple groups and singularities.
Also, Lehn, Namikawa and Sorger
http://lanl.arxiv.org/abs/1002.4107
give a classification of the nilpotent orbits for which the restriction of the adjoint quotient to the Slodowy (Kostant) slice gives a universal Poisson deformation of the central fibre.
I will edit this later, as I have to run and I am not entirely sure if this is the kind of statement you are looking for.