REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 3, Pages 444{456 (November 9, 1999) S 1088-4165(99)00066-7 THE ADJOINT REPRESENTATION OF A REDUCTIVE GROUP AND HYPERPLANE ARRANGEMENTS J. MATTHEW DOUGLASS Abstract. Let G be a connected reductive algebraic group with Lie algebra g defined over an algebraically closed field, k,withchark= 0. Fix a parabolic subgroup of G with Levi decomposition P = LU where U is the unipotent radical of P .Letu=Lie(U)andletzdenote 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,therootsoftin u determine an arrangement of hyperplanes in z we denote by z. In this paper we construct an isomorphism A P z z of graded k[z]-modules HomG(g∗;k[G (z+u)]) ∼= D( ), where D( )isthe z × A PA k[z]-module of derivations of . We also show that HomG(g∗;k[G (z+u)]) P A × z and k[z] HomG(g∗;k[G u]) are isomorphic graded k[z]-modules, so D( ) ⊗ ×P A and k[z] HomG(g∗;k[G u]) are isomorphic, graded k[z]-modules. It follows immediately⊗ that z is a× free hyperplane arrangement. This result has been proved using case-by-caseA 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 z. This result has also been proved, again using case-by-case× arguments, by SommersA and Trapa and independently by Broer. 1. Introduction Suppose k is an algebraically closed field with characteristic zero and G is a connected, reductive, algebraic group defined over k with Lie algebra g.Let denote the cone of nilpotent elements in g.ThenGacts on g by the adjointN representation and is a closed, G-invariant subvariety, so G acts on k[ ], the ring of regular functionsN on .Since is a cone, k[ ] inherits a gradingN from k[g] and each homogeneous component,N Nk[ ] ,isaG-stableN subspace. Kostant [5] N j proved that g occurs as a constituent of k[ ]j if and only if j is an exponent of the Weyl group of G. Precisely, he proved theN equality of polynomials in q: dim Hom (g,k[ ] )qj =qn1 + +qnl G N j ··· j 0 X≥ where n1,...,nl is the multiset of exponents of W (see [7, 6.2] for a definition of the exponents{ of}W ). § Sommers and Trapa [9] and also Broer [4] have generalized this result as follows. Fix a maximal torus, T , and a parabolic subgroup, P ,ofGwith T P .LetU denote the unipotent radical of P and let L be a Levi subgroup of P⊆ containing Received by the editors March 8, 1999 and, in revised form September 28, 1999. 1991 Mathematics Subject Classification. Primary 22E46. c 1999 American Mathematical Society 444 ADJOINT REPRESENTATION AND ARRANGEMENTS 445 T . We follow the convention that lower case Fraktur letters will denote the Lie algebra of the group denoted by the same upper case Roman letter, so for example t = Lie(T ), and u = Lie(U). Let z denote the center of l,soz t.Thenzacts on u by restricting the adjoint representation of g and the weights⊆ of z on u are the restrictions to z of the weights of t on u. The kernels of these weights determine a hyperplane arrangement in z we denote by (z, z). Orlik and Terao [8] have proven, using case-by-case arguments, that z is a freeA hyperplane arrangement and Orlik and Solomon [6] have computed theA exponents of z in all cases. Now P acts on u and so we can form the associatedA fibre bundle, G P u,over G/P with fibre u.ThenGacts on G P u by left multiplication and× so G acts on k[G P u], the ring of global regular× functions on G P u. Moreover, k[G P u] inherits× a grading from the scalar action of k on u, k[G× P u] k[G ×P u] , ∼= j 0 j and each homogeneous component is a G-stable subspace.× Sommers≥ and× Trapa [9], and independently Broer [4], have proven that g occurs asL a constituent of P z k[G u]j if and only if j is an exponent of . Precisely, they prove the equality of polynomials× in q: A P j nz nz (1.1) dim Hom (g,k[G u] )q =q 1 + +q r G × j ··· j 0 X≥ z z z where n1,...,nr is the multiset of exponents of as defined in [7, Definition 4.25]. { } A In the special case when P is a Borel subgroup of G,thenz=t, the exponents t P of are the exponents of the Weyl group of G,andk[ ]j and k[G u]j afford equivalentA representations of G for j 0, so we recover Kostant’sN result.× The proofs given by Sommers and≥ Trapa and by Broer of the equality (1.1) are both case-by-case arguments, with separate arguments for each type of root system. In this paper, our main result is that there is an isomorphism of graded k[z]- P z z modules Γ: HomG(g∗,k[G (z+u)]) D( ), where D( )isthek[z]-module of derivations of z. We also× show that there→ isA an isomorphismA of graded k[z]-modules A P P z between HomG(g∗,k[G (z+u)]) and k[z] HomG(g∗,k[G u]). Thus D( )and P× ⊗ × A k[z] HomG(g∗,k[G u]) are isomorphic, graded k[z]-modules, so by keeping track of the⊗ gradings involved,× and recalling that g affords a self-dual representation of G, we see that the isomorphism Γ explains the computation, (1.1), of Sommers, Trapa, and Broer. P z Our proof that HomG(g∗,k[G (z+u)]) and D( ) are isomorphic, graded k[z]- modules does not involve any case-by-case× computations.A We define Γ explicitly and then show, using that the set of semisimple elements is dense in g and some results on the P -orbit structure of z + u, that it’s a bijection. P z Since k[z] HomG(g∗,k[G u]) and D( ) are isomorphic k[z]-modules, it fol- lows immediately⊗ that the arrangement× z Ais free. We thus obtain a proof, free of case-by-case considerations, of the resultA of Orlik and Terao [8] that the restrictions of the reflection arrangements arising from Weyl groups are free. After learning of our ideas about the definition of Γ, Broer [3] has given another proof that it’s surjective, the difficult part of showing that it’s an isomorphism. His proof is elegant, using Saito’s criterion, but relies on a previous result of his concerning sums of exponents. Our proof that Γ is surjective is completely different than Broer’s and uses only some basic algebraic geometry and properties of root systems and algebraic groups. 446 J. MATTHEW DOUGLASS Moreover, our technique can be extended to give a new proof of a theorem of Broer’s [2, Theorem 1] that characterizes when the restriction mapping from T MorG(g,V)toMorW(t,V ) is an isomorphism for a rational G-module V .Ina subsequent paper we hope to give a parabolic analog of Broer’s theorem, where g is replaced by AdG(z + u) (the closure of a Dixmier sheet in g), t is replaced by z, and V T is replaced by V L. The rest of this paper is organized as follows. In 2 we collect several results about hyperplane arrangements in the form we need later;§ in 3 we discuss associ- ated fibre bundles over G/P ;in 4 we collect some results about§ the adjoint action of U on z + u;in 5 we construct§ Γ and prove it’s an isomorphism; and in 6we explain how the fact§ that Γ is an isomorphism implies the results discussed above.§ Finally, we use the following notation throughout this paper. If X and Y are varieties defined over k,thenk[X] denotes the k-algebra of global regular functions on X and Mor(X, Y ) denotes the set of all morphisms from X to Y . If a group, G, acts on X and Y ,thenMorG(X, Y ) denotes the set of all G-equivariant morphisms from X to Y .IfVand W are k-vector spaces, then V W and Hom(V,W)denote V W and Hom (V,W) respectively. If V and W ⊗afford representations of G, ⊗k k then HomG(V,W) denotes the set of all G-equivariant linear transformations from V to W . 2. Hyperplane arrangements In this paper a “hyperplane arrangement” is a pair, (V, ), where V is a finite dimensional k-vector space and is a finite multiset of hyperplanesA in V . This slight generalization of the usual meaningA has no effect on the invariants of arrangements we consider. Suppose (V, ) is an arrangement with dim V = n.Let µ 1 j m A {j| ≤ ≤ } be a multiset of linear functionals so that = ker µ1,...,ker µ . Fix a basis, A { m } e1,...,en ,ofV and let x1,...,xn be the dual basis of V ∗.Thenfor1 i m { } n { } ≤ ≤ we may write µj = i=1 ξi;j xi where the ξi;j ’s are in k.Wemayidentifyk[V] with the polynomial ring k[x1,...,xn]andif∂i denotes partial differentiation with P respect to xi,thenDerk[V], the k[V ]-module of k-linear derivations of k[V ], has basis ∂1,...,∂n .