Logarithmic Derivations of Adjoint Discriminants

We exhibit a relationship between projective duality and the sheaf of logarithmic vector fields along a reduced divisor D of projective space, in that the push-forward of the ideal sheaf of the conormal variety in the point-hyperplane incidence, twisted by the tautological ample line bundle is isomorphic to logarithmic differentials along D. Then we focus on the adjoint discriminant D of a simple Lie group with Lie algebra g over an algebraically closed field k of characteristic zero and study the logarithmic module Der_U(-log(D)) over U = k[g]. When g is simply laced, we show that this module has two direct summands: the G-invariant part, which is free with generators in degrees equal to the exponents of G, and the G-variant part, which is of projective dimension one, presented by the Jacobian matrix of the basic invariants of G and isomorphic to the image of the map ad : g ⊗ U(-1) → g ⊗ U given by the Lie bracket. When g is not simply laced, we give a length-one equivariant graded free resolution of Der_U(-log(D)) in terms of the exponents of G and of the quasi-minuscule representation of G