In this note we discuss an interesting duality known to occur for certain complex reflection groups, we prove in particular that this duality has a concrete representation theoretic realisation. As an application, we construct matrix factorisations of the highest degree basic invariant which give free resolutions of the module of K\"{a}hler differentials of the coinvariant algebra $A$ associated to such a reflection group. From this one can read off the Hilbert series of ${\rm Der}_{\mathbb{C}}(A,A)$. This applies for instance when $A$ is the cohomology of any complete flag manifold, and hence has geometric consequences. 查看全文>>