Chapter 9 Structure of reductive groups 9.1 First definitions and results 9.1.1 Examples Let us start to give some example of reductive and semisimple groups. Lemma 9.1.1 Let V be a faithful representation of G. Assume that V is a simple representation, then G is reductive. Ru Proof. Let Ru = Ru(G). Because Ru is normal the subspace V is a subrepresentation. It is non empty because Ru is unipotent therefore it is equal to V . In particular Ru acts trivially on V and because the representation is faithful Ru is trivial. Corollary 9.1.2 The groups GL(V ), SL(V ), SO(V ) and Sp(V ) (with dim V even in this last case) are reductive groups. Proof. In all cases, the standard representation V is faithful and simple. 9.1.2 Root datum Recall first the definition of a root system. Definition 9.1.3 A root system is a pair (R, V (R)) of a vector space V (R) and a finite set R such that the following conditions are satisfied. • 0 ∈ R and R spans V (R). • For any α ∈ R, there exists a linear form α∨ ∈ V (R)∨ such that α, α∨ = α∨(α) = 2 and ∨ sα ∈ Endk(V (R)) defined by sα(v)= v − v, α α preserves R. • For any α, β ∈ R we have α∨, β∈ Z. Definition 9.1.4 A root datum is a quadruple (M, M ∨, R, R∨) satisfying the following conditions: ∨ ∨ (ı) The sets M and M are free Z-modules of finite rank with M = HomZ(M, Z). We denote by m,f = f(m) for m ∈ M and f ∈ M ∨ the pairing. 85 86 CHAPTER 9. STRUCTURE OF REDUCTIVE GROUPS (ıı) The sets R and R∨ are finite subsets of M and M ∨ respectively with a bijection ∨ : R → R∨ denoted by α → α∨ such that the following conditions hold: α, α∨ = 2 and ∨ ∨ sα(R)= R, sα∨ (R )= R ∨ ∨ where σα(m)= m − m, α α and sα∨ (f)= f − α, fα . The root datum is called reduced if for α, β ∈ R, the condition Zα = Zβ ⇒ β = ±α is satisfied. Remark 9.1.5 (ı) Over algebraically closed fields, only reduced root data are useful. (ıı) A root is always a non zero element. ∨ ∨ ∨ (ııı) We shall denote by V and V the tensor products V = M ⊗ZR and V = M ⊗ZR. Then R is a root system in the subspace V (R) it spans in V and R∨ is the dual root system in V ∨(R∨) ≃ (V (R))∨. Definition 9.1.6 Let V be a vector space with R ⊂ V a root system, let R∨ be the dual root system in V ∨. (ı) We denote by Q(R) the subgroup of V spanned by R. This is a lattice and we call it the root lattice. We define in the same way the coroot lattice Q(R∨) in V ∨. (ıı) We denote by P (R) the subgroup of V defined by P (R)= {v ∈ V / v, α∨∈ Z, for all α∨ ∈ R∨}. This is called the weight lattice. We define in the same way P (R∨) which is the coweight lattice. Definition 9.1.7 The Weyl group of a root datum is the subgroup of GL(M ⊗Z R) generated by the reflections sα for α ∈ R. Definition 9.1.8 A root datum is called semisimple if Q(R) has the same rank as M (which is also the dimension of V ). Fact 9.1.9 The root datum is semisimple if and only if the dual root datum (M ∨, M, R∨,R) is semisimple i.e. if and only if rkQ(R∨) = rk(M ∨) Proof. If the root datum is semisimple, then V (R) = V and by properties of dual root systems, we have V ∨ = V ∨(R∨) which is therefore semisimple. Proposition 9.1.10 Let (M, M ∨, R, R∨) be a semisimple root datum. Then this datum equivalent to the root system (R, V ) together with the finite Z-submodule M/Q(R) of P (R)/Q(R). Proof. If we have a semisimple root datum then we have the inclusions Q(R) ⊂ M ⊂ P (R) and these Z-module have the same rank equal to dim V . In particular P (R)/Q(R) is finite. Conversely, if we have the root system R, then Q(R) and P (R) are well defined. Therefore if we have the finite submodule M/Q(R) we recover M by taking the inverse image of this module by the surjective map P (R) → P (R)/Q(R). 9.2. CENTRALISER OF SEMISIMPLE ELEMENTS 87 9.2 Centraliser of semisimple elements Theorem 9.2.1 Let G′ be an affine algebraic group and assume that G is a closed connected subgroup ′ ′ ′ of G . Let g = L(G) and let S be an abelian (not nec. closed) subgroup of G with S ⊂ Gs ∩ NG′ (G). Let GS = {g ∈ G / sgs−1 = g for all s ∈ S} gS = {η ∈ g / (Ad s)(η)= η for all s ∈ S}. Then we have L(GS)= gS. Let us first state a corollary of this result. Corollary 9.2.2 Let G be a connected algebraic group, let g = L(G) and let S be a closed diagonal- isable subgroup of G. Then we have the equality S g = L(CG(S)). Corollary 9.2.3 Let T be a maximal torus of a reductive group G act on g = L(G) via the adjoint representation. Let h = L(T ) then we have the equality h = gT . Proof. We start with the following proposition which deals with the case of a unique element. Proposition 9.2.4 Let G and G′ as above and let g and g′ be their Lie algebras. Let s be a semisimple element in G′ normalising G. Then we have the equality gs = L(Gs). Proof. Embedding G′ in some GL(V ), we may assume that G′ = GL(V ). We may also assume that s is a diagonal element s = diag(x1, · · · ,x1, · · · ,xr, · · · ,xr) where xi appears ni times. They we easily compute the following equalities r r ′ s ′ s ′ s (G ) = GLni and L((G ) )= glni = (g ) . Yi=1 Yi=1 Furthermore, Ad s is a semisimple element in gl(g′) thus we have a decomposition g′ = (g′)s ⊕ x where x is the direct sum of the eigenspaces of ad s with eigenvalue different from 1. Let Ad g s be the restriction of Ad s to g. Consider now the morphism φ : G′ → G′ defined by φ(x)= xsx−1s−1. On the one hand, its image X is a locally closed subset of G′ and it the translate by s−1 of the conjugacy class of s. Therefore dim X = dim(G′/(G′)s) = dim G′ − dim(G′)s. On the other hand, φ is the following composition Id×θ µ G′ / G′ / G′ where θ(x)= sx−1s−1. We have θ = (Ad s) ◦ i where i(x)= x−1 therefore we get deφ = Id+ deθ = Id − Ad s. ′ s ′ s In particular, we get that ker deφ = (g ) = L((G ) ). Therefore the image of deφ is of dimension ′ ′ s dim G − dim(G ) = dim X and thus deφ is surjective onto TeX. In particular we get TeX = x. s s s Let ψ = φ|G, then we have deψ = Idg − Adg s. But ψ is constant on G thus L(G ) ⊂ ker deψ = g . We are left to prove the inequality dim gs ≤ dim Gs. 88 CHAPTER 9. STRUCTURE OF REDUCTIVE GROUPS We proceed as before: let Y be the image of ψ. It is a locally closed subvariety in G of dimension s dim Y = dim G − dim G . Furthermore, since s normalises G, we have Y ⊂ G ∩ X thus TeY ⊂ g ∩ TeX = g ∩ x. Thus we have the inequality dim Y ≤ dim g ∩ x. On the other hand, we have the decomposition in (Ad s)-eigenspaces g′ = (g′)s ⊕ x which induces a decomposition g = gs ⊕ (g ∩ x) thus dim gs = dim g − dim(g ∩ x) ≤ dim g − dim Y = dim Gs. The result follows. S We prove the theorem by induction on dim G. If G = G , then Ad s = IdG and we are done. Otherwise, let s ∈ S with Gs proper in G. By the previous proposition we get that L((Gs)0) = gs. The group (Gs)0 is normalised by S (because S is abelian) thus by induction we have gS = (gs)S = L((Gs)0)S = L(((Gs)0)S) ⊂ L(GS). On the other hand, Int(s) acts trivially on GS thus its differential Ads acts trivially on L(GS ) and we get the converse inclusion L(GS ) ⊂ gS. 9.3 Structure theorem for reductive groups Let T be a maximal torus of G reductive and let it act on g via the adjoint representation. Then we have a decomposition in eigenspaces as follows: g = h ⊕ gα αM∈R where h = L(T ), where R is a subset of X(T ) \ {0} and where gα = {η ∈ g / t · η = α(t)η for all t ∈ T }. We want to prove the following result. Theorem 9.3.1 Let G be a reductive groupe and let T be a maximal torus. Let W (G, T ) be the Weyl group of G and g be the Lie algebra of G. Let h be the Lie algebra of T and R the set of non trivial characters of T appearing in g. Then we have gT = h and a decomposition g = h ⊕ gα αM∈R such that the following properties hold.