# Differential of the adjoint quotient map

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.

• There is no published follow-up, though there may well be some written notes somewhere. Roger was active in many directions, but died at the age of 63 in 1993 and left much work undone. (There has however been a lot of progress made since then in the study of nilpotent orbits, normality of their closures, etc.) Dec 2, 2015 at 23:49
• @JimHumphreys Thanks for providing more context. I am only partly aware of the works discussing the normality of closures. Richardson uses his results for rank($d\pi$) and a theorem of Borho-Kraft to check for normality. I was interested more directly in rank($d \pi$) and the methods employed by Richardson to compute it. He indicates some of the steps involved and mentions a relation to methods employed by Springer in a paper called "Regular elements of finite reflection groups". I was hoping to see more details of this calculation. Dec 3, 2015 at 16:49
• I can't suggest a helpful reference, but you might look at the papers listed on MathSciNet which cite Richardson's paper (including several by Panyushev). By the way, Springer's paper (which is important but not easy) is here: gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002091275 Dec 3, 2015 at 19:27