We discuss an interesting duality known to occur for certain
complex reflection groups, namely the duality groups. Our
main construction yields a concrete, representation theoretic
realisation of this duality. This allows us to naturally identify
invariant vector fields with vector fields on the orbit space,
for the action of a duality group. As another application,
we construct matrix factorisations of the highest degree
basic invariant which give free resolutions of the module of
Kähler differentials of the coinvariant algebra A associated
to such a reflection group. From this one can explicitly
calculate the dimension of each graded piece of ΩA/C and
of DerC(A, A), adding a new formula to the numerology of
reflection groups. This applies for instance when A is the
cohomology of any complete flag manifold, and hence has
geometric consequences.