The sheaf T<D> of logarithmic vector fields along a divisor D of the projective space P is the sheafification is a modification of the tangent bundle of P along D, controlling locally trivial deformations of D in P. I will discuss the structure of T<D> when D is the discriminant of a simple complex Lie algebra sitting in P=P(g), g being the Lie algebra of G. A particular emphasis will be on the case when D the Dynkin diagram of D is simply laced. The approach is based on projective duality and cohomology of homogeneous vector bundles. Joint work with Vladimiro Benedetti (Nice, France) and Simone Marchesi (Barcelona, Spain)