The adjoint representation of a reductive group and hyperplane arrangements

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