Lectures on Chevalley Groups

Abstract

We start with some basic properties of semisimple Lie algebras over C, and establish some notation to be used throughout. The assertions not proved here are proved in the standard books on Lie Algebras, e.g., those of Dynkin, Jacobson or Sophus Lie (Séminaire). Let L be a semisimple Lie algebra over C, and H a Cartan subalgebra of L. Then H is necessarily abelian and L = H ⊕ α =0 L α where α ∈ H * and L α = {X ∈ L : [H, X] = α(H)X, ∀ H ∈ H} Note that H = L 0. The α's are linear functions on H, called roots. We adapt the convention that L γ = 0 if γ is not a root. Then [L α , L β ] ⊆ L α+β. The rank of L equals dim C H = l, say. The roots generate H * as a vector space over C. Write V for H * Q , the vector space over Q generated by the roots. Then dim Q V = l. Let γ ∈ V. Since the Killing form is nondegenerate there exists an H γ ∈ H such that (H, H γ) = γ(H) for all H ∈ H. Define (γ, δ) = (H γ , H δ) for all γ, δ ∈ V. This is a symmetric, nondegenerate, positive bilinear form on V. Denote the collection of all roots by Σ. Then Σ is a subset of the nonzero elements of V satisfying: 0. Σ generates V as a vector space over Q. 1. α ∈ Σ ⇒ −α ∈ Σ and kα / ∈ Σ for k an integer = ±1. 2. 2(α, β)/(β, β) ∈ Z for all α, β ∈ Σ. (Write α, β = 2(α, β)/(β, β). These are called Cartan integers.) 3. Σ is invariant under all reflections w α (α ∈ Σ) (where w α is the reflection in the hyperplane orthogonal to α, i.e., w α v = v − 2(v, α)/(α, α)α). which has a Z-basis which is a C-basis for V , we say M is a lattice in V. We can now state the following corollaries to Theorem 2. Corollary 1. (a) Every finite dimensional L-module V contains a lattice M invariant under all X m α /m! (α ∈ Σ, m ∈ Z +); i.e., M is invariant under U Z. (b) Every such lattice is the direct sum of its weight components; in fact, every such additive group is. Proof. (a) By the theorem of complete reducibility of representations of semisimple Lie algebras over a field of characteristic 0 (See Jacobson, Lie Algebras, p. 79), we may assume that V is irreducible. Using Theorem 3, we find v + and set M = U − Z v +. M is finitely generated over Z since only finitely many monomials in U − Z fail to annhilate v +. Since U − v + = V and since U − Z spans U − over C, we see that M spans V over C. Before completing the proof of (a), we will first show that if c i v i = 0 with c i ∈ C, v i ∈ M and v 1 = 0, then there exist n i ∈ Z, n 1 = 0, such that c i n i = 0. To see this, let u ∈ U Z be such that the component of uv 1 in Cv + is nonzero. Then c i uv i = 0 implies c i n i = 0 where n i v + is the component of uv i in Cv +. We have n i ∈ Z by Lemma 12 and n 1 = 0 by choice of u. Finally, suppose a basis for M is not a basis for V. Let l be minimal such that there exist v 1 ,. .. , v l ∈ M linearly independent over Z but linearly dependent over C. Suppose l i=1 c i v i = 0. Then there exist n i ∈ Z, n 1 = 0 such that l i=1 c i n i = 0. We see that 0 = n 1 l i=1 c i v i = l i=2 c i (n 1 v i − n i v 1). Since n 1 v i − n i v 1 , i = 2, 3,. .. , 4 are linearly independent ...

Non-Archimedean Whittaker Functions as Characters: A Probabilistic Approach to the Shintani–Casselman–Shalika Formula

“…Let us give pointers to the proof. The two first claims are entirely contained in [Ste68] Corollary 4 p.26. For the third claim, existence and uniqueness of the factorisation (6.2) is Lemma 17 p.24 from [Ste68].…”

“…The two first claims are entirely contained in [Ste68] Corollary 4 p.26. For the third claim, existence and uniqueness of the factorisation (6.2) is Lemma 17 p.24 from [Ste68]. To see that the coefficients θ β (ht(β)=i) do not depend on the order, let us consider the product ht(β)=i x −β θ β ∈ N i for two different orders.…”

“…Let G be a split reductive group (scheme) over Z, as constructed by Chevalley (see for e.g [Ste68]). We fix a maximal torus T ≈ (G m ) r , where r is the rank of the group.…”

“…We will mainly work in the lower Borel subgroup B and its unipotent subgroup N . For every root β ∈ Φ, we have a group homomorphism x β : G a → G coming from a pinning and the group G is generated by the subgroups (x β (K)) (β∈Φ) ( §3 in [Ste68]). We fix this pinning throughout the paper.…”

Generation of pairs of short root subgroups in Chevalley groups

“…(see [St,Lecture 38]), and any element of H(C 2 ) has the form h 1−2 (k 1 )h 2 (k 2 ), k i ∈ K * , i = 1, 2. Therefore, the condition hx ρ (t)x δ (ct)h −1 = x ρ (t )x δ (c t ) amounts to the following system of equations:…”