Let G G be a connected reductive algebraic group with Lie algebra g \mathfrak g defined over an algebraically closed field, k k , with char k = 0 \operatorname {char} k=0 . Fix a parabolic subgroup of G G with Levi decomposition P = L U P=LU where U U is the unipotent radical of P P . Let u = Lie ( U ) \mathfrak u=\operatorname {Lie}(U) and let z \mathfrak z denote the center of Lie ( L ) \operatorname {Lie}(L) . Let T T be a maximal torus in L L with Lie algebra t \mathfrak t . Then the root system of ( g , t ) (\mathfrak g, \mathfrak t) is a subset of t ∗ \mathfrak t^* and by restriction to z \mathfrak z , the roots of t \mathfrak t in u \mathfrak u determine an arrangement of hyperplanes in z \mathfrak z we denote by A z \mathcal A^{\mathfrak z} . In this paper we construct an isomorphism of graded k [ z ] k[\mathfrak z] -modules Hom G ( g ∗ , k [ G × P ( z + u ) ] ) ≅ D ( A z ) \operatorname {Hom}_G(\mathfrak g^*, k[{G\times ^P(\mathfrak z+\mathfrak u)}]) \cong D(\mathcal A^{\mathfrak z}) , where D ( A z ) D(\mathcal A^{\mathfrak z}) is the k [ z ] k[\mathfrak z] -module of derivations of A z \mathcal A^{\mathfrak z} . We also show that Hom G ( g ∗ , k [ G × P ( z + u ) ] ) \operatorname {Hom}_G(\mathfrak g^*, k[{G\times ^P(\mathfrak z+\mathfrak u)}]) and k [ z ] ⊗ Hom G ( g ∗ , k [ G × P u ] ) k[\mathfrak z] \otimes \operatorname {Hom}_G(\mathfrak g^*, k[G \times ^P \mathfrak u]) are isomorphic graded k [ z ] k[\mathfrak z] -modules, so D ( A z ) D(\mathcal A^{\mathfrak z}) and k [ z ] ⊗ Hom G ( g ∗ , k [ G × P u ] ) k[\mathfrak z] \otimes \operatorname {Hom}_G(\mathfrak g^*, k[G \times ^P \mathfrak u]) are isomorphic, graded k [ z ] k[\mathfrak z] -modules. It follows immediately that A z \mathcal A^{\mathfrak z} is a free hyperplane arrangement. This result has been proved using case-by-case arguments by Orlik and Terao. By keeping track of the gradings involved, and recalling that g \mathfrak g affords a self-dual representation of G G , we recover a result of Sommers, Trapa, and Broer which states that the degrees in which the adjoint representation of G G occurs as a constituent of the graded, rational G G -module k [ G × P u ] k[G\times ^P \mathfrak u] are the exponents of A z \mathcal A^{\mathfrak z} . This result has also been proved, again using case-by-case arguments, by Sommers and Trapa and independently by Broer.