There is a well-known correspondence between simple singularities and (finite, simply laced) Coxeter groups, where the monodromy of the singularity is given by the Coxeter element. Traces of this analogy reach the generality of well-generated groups, the largest class of complex reflection groups that have good analogs of Coxeter elements. In that setting, Bessis generalized the function-theoretic Lyashko-Looijenga morphism to a map (LL) that describes the discriminant hypersurface H of W, via a ramified covering of a points configuration space.

There is a natural bijective correspondence (``Trivialization Theorem’’) between points in a generic fiber of the LL-map, and reduced reflection factorizations of a Coxeter element c of W. This is fundamental in Bessis’ work, where the combinatorics of reduced factorizations is used as a recipe to construct the universal covering space of the complement W \ (V - H). However, it relies on an elusive numerological coincidence between the degree of the LL-map and the number of factorizations, that has only been explained in the generality of Weyl groups after work of Jean Michel.

We will review various structural properties of the LL map (and, where necessary, develop new ones), produce some finer enumerative results, and propose a uniform approach towards the proof of the Trivialization Theorem.’’