Krammer representations for complex braid groups

Let B be the generalized braid group associated to some finite complex reflection group W. We define a representation of B of dimension the number of reflections of the corresponding reflection group, which generalizes the Krammer representation of the classical braid groups, and is thus a good candidate in view of proving the linearity of B. We decompose this representation in irreducible components and compute its Zariski closure, as well as its restriction to parabolic subgroups. We prove that it is faithful when W is a Coxeter group of type ADE and odd dihedral types, and conjecture its faithfulness when W has a single class of reflections. If true, this conjecture would imply various group-theoretic properties for these groups, that we prove separately to be true for the other groups.