Indian J. Pure Appl. Math., 46(5): 695-700, October 2015 °c Indian National Science Academy DOI: 10.1007/s13226-015-0134-7

ON CHEVALLEY’S Z-FORM

M. S. Raghunathan

National Centre for , Indian Institute of Technology, Mumbai 400 076, India e-mail: [email protected]

(Received 28 October 2014; accepted 18 December 2014)

We give a new proof of the existence of a Chevalley basis for a semisimple over C.

Key words : Chevalley basis; semisimple Lie algebras.

1. INTRODUCTION

Let g be a complex . In a path-breaking paper [2], Chevalley exhibitted a basis B of g (known as Chevalley basis since then) such that the structural constants of g with respect to the basis B are integers. The basis moreover consists of a basis of a Cartan subalgebra t (on which the roots take integral values) together with root vectors of g with respect to t. The structural constants were determined explicitly (up to signs) in terms of the structure of the of g. Tits [3] provided a more elegant approach to obtain Chevalley’s results which essentially exploited geometric properties of the root system. Casselman [1] used the methods of Tits to extend the Chevalley theorem to the Kac-Moody case. In this note we prove Chevalley’s theorem through an approach different from those of Chevalley and Tits.

Let G be the simply connected algebraic corresponding to g. Let T be a maximal torus (the Lie subalgebra t corresponding to it is a Cartan subalgebra of G). Let Φ be the root system of G with respect to T and ∆ a simple root system. For ϕ ∈ Φ, let G(ϕ) be the 3-dimensional subgroup (isomorphic to SL(2)) corresponding to ϕ. For an element v 6= 0 in the root space gϕ, there is a unique natural isomorphism fϕ,v of SL(2) on G(ϕ) taking the diagonal group into T and the induced Lie algebra map taking e12 to v. Set f(ϕ,v)(e12 − e21) = sϕ,v. The key result we prove in this note is the following. 696 M. S. RAGHUNATHAN

α ˜ For each α ∈ ∆, let eα be a non-zero element in g and set sα = sα,eα . Let W be the sugroup of G generated by {sα|α ∈ ∆}. It is well known that W˜ ⊂ N(T), the normalizer of T. Then W˜ ∩ T = T(2), the group of 2-torsion elements in T. Further if w, w0 ∈ W˜ and α, α0 ∈ ∆ with 0 0 w(α) = w(α ), Ad(w)(eα) = ±Ad(w )(eα0 ).

One deduces then the existence of a Chevalley basis from the above two assertions. The second assertion above is in fact equivalent to the existence of the Chevalley basis and the first can be deduced from the existence. The first assertion seems to be of some interest in itself. It says that while the sequence {1} → T → N(T) → N(T)/T → {1} does not split it comes close to to doing so.

2. REPRESENTATIONS OF SL(2, C)

We recall in this section some basic facts about finite dimensional (holomorphic) representations of the SL(2, C) (or simply SL(2)), the group of (2 × 2)-matrices over C of determinant 1 (under multiplication). D will denote the group of diagonal matrices in SL(2). For t ∈ C∗, d(t) denotes −1 2 the diagonal matrix with d(t)11 = t and d(t)22 = t . We set C = V and denote by ρ the natural representation of SL(2) on V . We denote by sl(2, C) (or simply sl(2)), the Lie algebra {X²M(2, C)| trace(X) = 0} of SL(2). We let ρ˙ denote the natural representation of sl(2) on V (induced by

ρ). For 1 ≤ i, j ≤ 2, let eij be the matrix whose (k l)-th entry is δik · δjl. Set s = e12 − e21, e+ = e12, e− = e21 and h = e11 − e22. Then s ∈ SL(2) is the “Weyl” element; it normalizes 2 D and s = −(identity). Further, {e+, h, e−} is a basis of sl(2); and we have [h, e+] = 2 · e+,

[h, e−] = −2 · e− and [e+, e−] = h.

For an integer m ≥ 0 we denote by ρm (resp. ρ˙m) the representation of SL(2) (resp. sl(2)) induced by ρ on the symmetric m-th power Sm(V ) of V . If {u, v} is the standard basis of V = C2, p q m ∗ {u · v | p + q = m} is a basis of S (V ). Let t²C and s²SL(2) be the matrix e+ − e−. One then has, as is easily seen by inspection:

Lemma 2.1 — p q p−q p q ρm(d(t))(u · v ) = t · u · v

p q p+1 q−1 ρ˙m(e−)(u · v ) = q · u · v

p q p−1 q+1 ρ˙m(e+)(u · v ) = p · u · v

p q p q ρ˙m(h)(u · v ) = 2 · (p − q) · u · v ON CHEVALLEY’S Z-FORM 697

p q p q p p ±1 ±(p−q) p q ρm(s)(u · v ) = (−1) · u · v = (−1) · (q!/p!) · ρ˙m(e∓) (u · v ) according as p ≥ q or q ≥ p.

For any integer m ≥ 0, ρm is an irreducible representation of SL(2) of dimension m + 1 and, up to equivalence, it is the unique irreducible representation of dimension m + 1. Moreover every representation τ of SL(2) is a direct sum of irreducible representations; thus τ is a direct sum of copies of the “ρm”s. One deduces from the lemma above the following:

Corollary 2.2 — Let τ be an irreducible representation of SL(2) on a vector space E of dimension m + 1. If z is an eigen vector for τ(h) in E with eigen value k,

(m+k)/2 k τ(s)(z) = (−1) · ((m − k)/2)! · τ(e∓) (z)/((m + k)/2)!

In particular, if z²E is a D-fixed vector, τ(s)(z) = ±z.

3. SEMISIMPLE LIE GROUPS,MAXIMAL TORI,ROOTS

3.1 Maximal torus, roots and root-spaces

Let G be a connected simply connected semisimple over C and g its Lie algebra. Let T be a maximal torus in G and t ⊂ g the Lie subalgebra of g corresponding to t. A root of G (with respect to T) is a non-trivial eigen character ϕ : T → C∗ for the adjoint action of T on the Lie algebra g. We denote by X(T) the group of characters on T and by Φ ⊂ X(T) the set of roots. For ϕ ∈ Φ, gϕ = {v ∈ g|Ad(t)(v) = ϕ(t) · v, ∀t ∈ T}, the eigenspace corresponding to ϕ, “the root- ` ϕ space of ϕ” is of dimension 1 and g = t ⊕ ϕ∈Φ g . The Killing form < ., . > on g, restricted to t is ∗ ∗ −1 non-degenerate. Hence it defines an isomorphism A of t on its dual t . For λ ∈ t , set A (λ) = Hλ. ∗ ∗ We have then for H ∈ t and λ ∈ t , < Hλ, H > = λ(H). For λ, µ ∈ t , set < λ, µ >=< Hλ,Hµ >.

3.2 Simple roots; the

Let N(T) be the normalizer of T (in G) and W = N(T)/T, the Weyl group. The inner conjugation action of N(T) on T factors through to an action of W on T and hence on X(T); the set Φ is stable under this action. The set {< ϕ, ϕ >1/2 | ϕ ∈ Φ} of “root-lengths” has at most two elements and the Weyl group acts transitively on the set of all roots of the same length. We fix a lexicographic (total) order in X(T) and denote by Φ+ (resp. Φ−, resp. ∆) the system of positive (resp. negative, resp. simple) roots. Let Ω be the set of roots dominant for the simple system ∆. The cardinality of Ω is the same as that of the set of root-lengths. For β ∈ Ω, Pβ is the parabolic subgroup determined by β: it is the sugroup of G corresponding to the Lie sub-algebra generated by {gϕ|ϕ ∈ Φ+}, t and {gϕ| ϕ ∈ Φ, < ϕ, β >= 0}. 698 M. S. RAGHUNATHAN

3.3 The 3-dimensional sub-algebras gα

α −α For α in Φ, let hα = 2 · Hα/ < Hα,Hα >. Now for any 0 6= v+ ∈ g ; there is a unique v− ∈ g with [v+, v−] = hα. Define iα,v : sl(2) → g by setting iα,v(e±) = v±, iα,v(h) = hα: it is a Lie α −α algebra isomorphism of sl(2) on the Lie subalgebra g ⊕ C · hα ⊕ g = g(α); the Lie subgroup

G(α) corresponding to g(α) is an algebraic subgroup and iα,v integrates to an isomorphism fα,v of GL(2) on G(α) - we have assumed that G is simply connected. We set sα,v = fα,v(s). As 2 2 s = −(Identity) in SL(2), sα,v is the unique non-trivial central element in G(α) and is of order 2.

It is easy to see that sα,v belongs to N(T) and that it acts trivially on {t ∈ T|α(t) = 1}, the kernel of α.

3.4 The group W˜

For each α in ∆ we now fix elements e±α with [eα, e−α] = hα and set iα,eα = iα, fα,eα = fα and ˜ ˜ ˜ sα,eα = sα. Let W be the subgroup of G generated by S = {sα|α ∈ ∆}. Then W maps onto W under the projection map π : N(T) → W. Let wα = π(sα) and S = π(S˜). Then (W, S) is a Coxeter system. Let B be the kernel of π : W˜ → W; B = T ∩ W˜ . We have with this notation the first key result.

Proposition 3.5 — B = T2, the subgroup of 2-torsion elements in T.

PROOF : Since (W, S) is a Coxeter system the group B is generated as a normal subgroup by p {(sα · sβ) αβ |α, β ∈ ∆}: here pαβ is the order of wα · wβ in W. It suffices therefore to prove that

pαβ 2 {(sα · sβ) |α, β ∈ ∆} are all of order 2: note that when α = β, pαβ = 1 and the {sα|α ∈ ∆} generate T(2) since G is simply connected. We will deal with each of the cases p = pαβ = 1, 2, 3, 4 and 6 separately:

2 1. p = 1. This is the case α = β and by its very definition sα has order 2.

2. p = 2. In this case the groups G(α) and G(β) commute, hence so do sα and sβ; it follows that 2 (sα · sβ) has order 2.

3. p = 3. Here gα ⊕ g(α+β) (resp. gβ ⊕ g(α+β)) is an irreducible G(β)- (resp. G(α)-) module

(of dimension 2). One sees now (from of SL(2)) that we have sβ(eα) = (α+β) [eβ, eα] = −[eα, eβ] = −sα(eβ). Set v = [eβ, eα] ∈ g . It follows that sβ · sα · sβ = −1 2 −1 sβ · sα · sβ · sβ = s(α+β),v(mod T(2)) = s(α+β),v(mod T(2)) = sα · sβ · sα(mod T(2)); as −1 2 sα = sα ( mod T(2)), (sα · sβ · sα) ∈ T(2).

4. p = 4. The root system is necessarily of type B2. We take α to be the long root. We then have ±α ±(α+β) ±(α⊕2·β) sα(β) = α + β and < β, α + β >= 0. The subspace g ⊕ g ⊕ g of g is ON CHEVALLEY’S Z-FORM 699

then an irreducible G(β)(' SL(2))-representation and one concludes from Corollary 2.2 that ±(α+β) Ad(sβ)(v) = v for all v ∈ g . It follows that sβ and s((α+β),v) commute for all v 6= 0 in α+β −1 2 g . Now sα · sβ · sα = sα · sβ · sα (mod T(2)) = s((α+β),sα(eβ )). Thus (sα · sβ) ∈ T(2) 4 and hence (sα · sβ) = 1.

5. p = 6. The root system is of type . We take α to be the long root. Then {m · α + n · β | 0 ≤

m ≤ 2, 0 ≤ n ≤ 3, m + n 6= 0} is the set of positive roots. Now sβ (resp. sα) conjugates

G(α) (resp. G(β)) into G(α + 3 · β) (resp. G(β + α)) taking sα (resp. sβ) into an element η (resp. ξ) with η2 (resp. ξ2) in T(2). As (α + 3 · β) ± (α + β) are not roots, G(α + 3 · β) and 3 2 2 G(β + α) commute. Thus (sα · sβ) = sα · sβ · sα · sβ · sα · sβ = ξ · sα · η · sβ = ξ · η (mod 6 2 2 T(2)). Hence (sα · sβ) = (ξ) · (η) (mod T(2)) and thus is in T(2).

This completes the proof of Proposition 3.5.

4. THE CHEVALLEY BASIS The key step we need is:

0 0 0 0 Proposition 4.1 — If α, α ∈ Φ and w, w ∈ W˜ are such that w(α) = w (α ), then Ad(w)(eα) 0 = ±Ad(w )(eα0 ).

0 0 −1 0 0−1 0 PROOF : Set β = w(α) = w (α ). Then one has w·sα ·w = s(β,Ad(w)(eα)) and w ·sα ·w = 0 0 s 0 0 . We denote these two elements by τ and τ respectively. Then τ, τ ∈ G(β). As they (β,Ad(w )(eα)) 0 both have the same wβ in W, we have τ = τ · θ with θ ∈ G(β) ∩ T(2); it follows that θ is β ±(Identity) in G(β). The proposition now follows from the fact that the map v → sβ,v of g \{0} into W˜ is injective.

Definition 4.2 — A basis B for g is adapted to a maximal torus T if the following conditions hold.

1. For some simple system ∆ in Φ, the root system of G with respect to T, B ∩ t = {Hα|α ∈ ∆} ϕ 2. For any root ϕ ∈ Φ, B ∩ (g \{0}) is non-empty =: {eϕ}.

3. For α in ∆, [hα, e±α] = ±2 · eα and [eα, e−α] = hα].

4. If fα : SL(2) → G(α) is the isomorphism given by fα(±e) = e±α and fα(h) = hα and

W˜ is the group generated by the {sα|α ∈ ∆}, then for ϕ ∈ Φ and w ∈ W˜ , Ad(w)(eϕ) =

²(w, ϕ) · ew(ϕ) where ²(w, ϕ) = ²(w, −ϕ) = ±1.

Remark 4.3 :

1. The discussion preceding the definition shows that a basis adapted to a maximal torus exists. 700 M. S. RAGHUNATHAN

2. In view of the fact that every root has a W˜ -conjugate in ∆, we have for any ϕ ∈ Φ, [hϕ, e±ϕ]

= 2 · e±ϕ and [eϕ, e−ϕ] = hϕ. ϕ 3. If {hα| α ∈ ∆} ∪ {eϕ ∈ g | ϕ ∈ Φ} is a basis adapted to T and δ : Φ → ±1 is a map with

δ(ϕ) = δ(−ϕ) for all ϕ ∈ Φ, {hα| α ∈ ∆} ∪ {δ(ϕ) · eϕ | ϕ ∈ Φ} is again a basis adapted to T. 4. One can find a basis adapted to a maximal torus such that for any automorphism σ of the

Dynkin diagram of ∆, σ(eα) = eσ(α).

5. For ϕ ∈ Φ, the map iϕ : sl(2) → g defined by setting iϕ(e±) = e±ϕ, iϕ(h) = hϕ is an iso-

morphism of sl(2) on g(ϕ), the Lie algebra of G(ϕ); if fϕ : SL(2) → G(ϕ) is the morphism ˜ −1 induced by iϕ and sϕ = fϕ(s), then for w ∈ W and ϕ ∈ Φ, w · sϕ · w = sw(ϕ) (mod T(2));

as T(2) ⊂ W˜ and every ϕ in Φ is in the W˜ orbit of an α ∈ ∆, sϕ is in W˜ .

Chevalley’s theorem — We will now establish the following result which asserts that a basis adapted to a maximal torus is indeed a Chevalley basis.

Theorem 4.4 — Let hα|α ∈ ∆} ∪ {eϕ|ϕ ∈ Φ} be a basis adapted to a maximal torus of g. Then for ϕ, ψ, ϕ + ψ in Φ, [eϕ, eψ] = ±Nϕ,ψe(ϕ+ψ) where Nϕ,ψ − 1 = max{k| ϕ − k · ψ ∈ Φ}. ˜ −1 PROOF : We begin with the observation that for w ∈ W and ϕ ∈ Φ, w · sϕ · w = sw(ϕ) (mod T(2)). Let ϕ, ψ and ϕ + ψ be in Φ. Let m be the subalgebra of g (of rank 2) generated by {gk·ϕ+l·ψ| k, l ∈ Z, (k · ϕ + l · ψ) ∈ Φ}. Then m is a semi-simple Lie algebra of rank 2. In the light of Proposition 4.1, we see now that it suffices to prove the theorem for groups of rank 2. Thus we assume that g = m and h = C · hϕ ⊕ C · hψ. Let h ∈ h be an element such that ϕ(h) = 0 while

ψ(h) > 0. We take on the dual of hR = R · hϕ ⊕ R · hψ, the lexiographic order determined by the + ordered basis (h, hφ). Then one sees easily that ϕ is a simple root i.e, φ ∈ ∆ while ψ ∈ Φ . Let E ϕ ` (ϕ+r·ψ) be the smallest Ad(G(ψ))-stable subspace of g containing g . Then E = r g is easily seen to be an irreducible representation of G(ψ)(' SL(2)). The desired results now follow Lemma 2.2. Since ϕ + ψ is a root we have only three possibilities: (i) ϕ − ψ is not a root or (ii) ϕ − ψ is a root, but ϕ − 2 · ψ is not a root or (iii) ϕ − ψ as well as ϕ − 2 · ψ are roots root. Correspondingly the dimension of E is 2, 3 or 4.

REFERENCES

1. Bill Casselman, Structure constants of Kac-Moody Lie algebras, Preprint (2013). 2. Claude Chevalley, Sur certains groupes simple, Tohuku Mathematics Journal, 48 (1955), 14-66. 3. Jacques Tits, sur les constants de structure et le theoreme d’existance des algebre de Lie semi-simple, Publications de l’I.H.E.S., 31 (1966), 21-58.