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I have often seen in the literature the statement that the orbit spaces of irreducible finite Coxeter groups are equivalent to unfoldings of singularities.

For instance, taken from Dubrovin, Differential geometry of the space of orbits of a finite Coxeter group:

(*) According to this the complexified orbit space of an irreducible Coxeter group is bi-holomorphic equivalent to the universal unfolding of a simple singularity.

For the crystallographic Coxeter groups, the proof of this seems scattered over the literature, with Arnold's books on singularity theory the main reference.

I have not been able to find a reference which proves this for non-crystallographic Coxeter groups.

Question: Is (*) true or false for non-crystallographic Coxeter groups? If it is true, what is a reference?

The best I have been able to find is the paper of Scherbak (Wavefronts and reflection groups) which embeds the discriminants of $H_2$, $H_3$ and $H_4$ in the discriminants of the groups $A_4$, $D_6$ and $E_8$ respectively. However, this is not exactly the same statement as (*).

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  • $\begingroup$ Arnold's group did pursue the non-crystallographic finite Coxeter groups after dealing with the crystallogaphic ones (ADE type and others). But the literature gets complicated to track, in part because his student Oleg Shcherbak didn't live to complete his Ph.D. There are also some small notes in Russian around that time by Arnold himself which didn't get translated (in Uspehi Mat. Nauk, where initially only longer articles got translated in Russian Math. Surveys). I think they did treat all finite Coxeter groups in much the same style. Maybe try the M. Davis book? $\endgroup$
    – Jim Humphreys
    Commented Jun 11, 2016 at 22:46

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