My question is regarding a paper by R.W Richardson titled "Derivatives of invariant polynomials on a semisimple Lie Algebra" ** . In this paper, he reports on computations of the rank of the differential of the adjoint quotient map $\pi : g \rightarrow h/W$, where $g$ is a complex semi-simple Lie algebra of rank $n$. Denote this differential by $d \pi_x$, where $x \in g$. The case where $x$ is nilpotent is most interesting and for the case of $x$ being regular nilpotent, we have rank($d \pi_x$)=n (a result that goes back to Kostant). Richardson's work is a generalization of this for other nilpotents. Now, in the paper, he alludes to a detailed exposition of results that were to appear elsewhere. But, I have not been able to track down this other work. Is anyone aware of this more detailed exposition ?
** Full Reference : Richardson, R. W. "Derivatives of invariant polynomials on a semisimple Lie algebra." Miniconference on Harmonic Analysis. Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1987 , Available here.