The Adjoint Representation of a Reductive Group and Hyperplane Arrangements

Let G be a connected reductive algebraic group with Lie algebra g defined over an algebraically closed field, k, with char k = 0. Fix a parabolic subgroup of G with Levi decomposition P = LU where U is the unipotent radical of P . Let u = Lie(U) and let z denote the center of Lie(L). Let T be a maximal torus in L with Lie algebra t. Then the root system of (g, t) is a subset of t∗ and by restriction to z, the roots of t in u determine an arrangement of hyperplanes in z we denote by Az. In this paper we construct an isomorphism of graded k[z]-modules HomG(g ∗, k[G×P (z + u)]) ∼= D(Az), where D(Az) is the k[z]-module of derivations of Az. We also show that HomG(g∗, k[G×P (z + u)]) and k[z]⊗HomG(g∗, k[G×P u]) are isomorphic graded k[z]-modules, so D(Az) and k[z]⊗HomG(g∗, k[G×P u]) are isomorphic, graded k[z]-modules. It follows immediately that Az 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 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×P u] are the exponents of Az. This result has also been proved, again using case-by-case arguments, by Sommers and Trapa and independently by Broer.