  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 $\mathfrak{g}$ over an
algebraically closed field $\mathbf{k}$ of characteristic zero and study the
logarithmic module Der$_{\mathbf{U}}(-\log(D))$ over
$\mathbf{U}=\mathbf{k}[\mathfrak{g}]$. When $\mathfrak{g}$ 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 $\mathbf{ad} : \mathfrak{g} \otimes \mathbf{U}(-1) \to \mathfrak{g}
\otimes \mathbf{U}$ given by the Lie bracket. When $\mathfrak{g}$ is not simply
laced, we give a length-one equivariant graded free resolution of
Der$_{\mathbf{U}}(-\log(D))$ in terms of the exponents of $G$ and of the
quasi-minuscule representation of $G$.
