12 Roots and Coroots

IfGis a semisimple algebraic group, there are two root systems Φ and ˆΦ associated withG, which are in duality: there is a bijectionα −→αˆ from the roots in Φ to the coroots in ˆΦ. The Weyl groups W are isomorphic, but long roots in Φ correspond to short roots in ˆΦ.

For many purposes it is useful to put the roots and coroots in the same vector space V, so that we can write (as we have been writing)

α^∨ = 2α

hα, αi. (54)

However Φ and ˆΦ arise differently, and in this section we will describe them as living in different vector spaces, V and its dual space V^∗. The reason we may put them together is that the ambient space V of Φ has a W-invariant inner product, so we may identify it withV^∗. But let us keep them separated for the moment.

LetT be a maximal torus ofG, which we will assume to be defined and split over F. Thus T ∼= G^r_m for some r, the rank of G. We will denote by X∗(T) ∼= Z^r the group of one-parameter subgroups, that is, algebraic homomorphismsG_m−→T, and by X^∗(T) ∼= Z^r the group of rational characters, that is, algebraic homomorphisms T −→G_m. There is a dual pairing between these groups, since given a one parameter subgroup i : G_m −→ T and a character λ : T −→ G_m, the composition is an endomorphism of G_m of the form x 7→ x^k, so (i, λ) −→ k gives a pairing X∗(T)× X^∗(T)−→Z. We will denote this by h, i.

Let V = R⊗X^∗(T) and V^∗ = R⊗X_∗(T). The roots Φ will live in V, and the coroots ˆΦ will live inV^∗. The roots, as we have mentioned, are the nonzero elements of X^∗(T) that occur in the adjoint representation Ad : G −→ End(g), where g is the Lie algebra of G. The coroots are elements of the dual module of X^∗(T) that implement the simple reflections. That is, we have for every rootαa simple reflection s_α :X^∗(T)−→X^∗(T), and there is a unique element α^∨ of the dual module X∗(T) of X^∗(T) such that for λ ⊂X^∗(T) we have

s_α(λ) =λ− hλ, α^∨iα.

Then α^∨ is a coroot.

IfGandG⁰are connected algebraic groups, anisogenyis a morphismf :G−→G⁰ that is a finite covering map. The mapf will be surjective in the sense of algebraic groups. However the induced map G(F) −→ G⁰(F) may not be surjective if the ground field F is not algebraically closed. For example if F is a finite field, G(F) and G⁰(F) will have the same number of elements. Given g⁰ ∈ G⁰(F) there exists

g ∈ G( ¯F) such that f(g) = g⁰, but g may not be rational over F. For semisimple algebraic groups, the kernel of an isogeny f will always be a subgroup of the finite center.

The semisimple group G is simply-connected if there are no nontrivial isogenies G⁰ −→G. If the ground field is C, this is equivalent toG(C) being simply-connected in the topological sense. We sayGisadjoint-type if there are no nontrivial isogenies G−→G⁰. This means that the center of G is trivial.

Given a Cartan type, there is a unique simply connected group in the correspond-ing isogeny class, which we will denoteG_sc. It has a finite center, and we will denote G_sc/Z(G_sc) asG_ad, theadjoint form. The groupπ₁(G_ad)∼=Z(G_sc) is a finite abelian group. It may be shown that it is isomorphic to P/Q where Q is the root lattice (spanned by the roots in V) and P is the weight lattice (consisting of λ ∈ V such that hλ, α^∨i ∈ Z for all corootsα^∨). It is given in the following table for the simple Cartan types.

Cartan Type π₁(G_ad)

A_r Z_r+1

B_r Z₂

C_r Z₂

D_r

Z₄ r odd Z₂×Z₂ r even

E₆ Z₃

E₇ Z₂

E₈ 1

F₄ 1

G₂ 1

(Zn=Z/nZ)

Table 5: Fundamental groups of simple groups of Lie type.

Let T_sc and T_ad be split maximal tori in G_sc and G_ad, arranged so the isogeny G_sc −→G_ad maps T_sc −→T_ad. The fundamental group of G_ad is the center of G_sc. We have maps:

X∗(T_sc) X^∗(T_sc)

↓ ↑

X∗(Tad) X^∗(Tad)

(55) They are of course all injective. It is useful to bear in mind that the roots span X^∗(T_ad) and that the coroots span X_∗(T_sc).

It is useful to have embeddings SL₂ −→Gwhich have good integrality properties.

One way to do this is to realize the Lie algebrag_F =F⊗_Zg_Zwhereg_Zis a Lie algebra defined over Z. It was proved by Chevalley, Sur certains groupes simples, T ˜Ahoku Math. J. (2), 7:14–66, 1955, that every semisimple complex Lie algebra g_C had such a basis. By tensoring this with an arbitrary field F, one obtains a Lie algebra g_F, and the Lie algebras of semisimple split reductive groups can all be obtained this way.

To give a bit more detail, the semisimple Lie algebra g=g_F decomposes via the adjoint representation as

g=t⊕M

α∈Φ

g_α,

wheretis the Lie algebra of T and g_α is the root eigenspace. Chevalley showed that we may choose elementsX_α ofg_α such that [X_α, X_β] =±(p+ 1)X_α+β whenα, β ∈Φ are such that α+β is a root, where p is the greatest integer such that β−pα ∈Φ.

Moreover, if H_α = [X_α, X_−α] then H_α ∈ t, and [H_α, X_β] = hβ, α^∨iX_β. Thus the H_α ∈ t themselves form a root system isomorphic to ˆΦ, and sometimes the H_α are called coroots, though we will use that term differently.

The group G_ad may then be taken to be the group of inner automorphisms ofg.

The group G_sc may be taken to be the universal covering group of G_ad. For every α∈Φ, the elementsXα, X−α and Hα satisfy

[H_α, X_α] = 2X_α, [H_α, X−α] =−2X−α, [X_α, X−α] =H_α.

These are the defining relations of the SL₂ Lie algebra. Since SL₂ is simply connected, it follows that there is a homomorphism i_α : SL₂ −→ G such that the induced map di_α on Lie algebras that satisfies

di_α

It would be correct but potentially confusing to write this asα^∨(t) so we will write instead

We will also make use of homomorphismx_α :F −→G(F) defined for α ∈Φ by Proof Equation (58) is an exponentiated version of the formula

Ad(h_α^∨(t))X_β =t^hα∨,βiX_β,

which is what we will verify. Since Xβ spans a root eigenspace and iα

t t⁻¹

there is some integer k such that Ad we remember how to pass from an action of the Lie group to the Lie algebra: we differentiate and set t = 0. In other words, if ρ : G −→ GL(V) is a representation

Let us consider an example. Let G= Sp₄. This group is simply-connected. It is the group of g ∈SL4 such that

Then g is the Lie algebra of matrices of trace 0 satisfyingXJ+J^tX = 0. Let T be

The differentials of these maps send the unit vector to H_α1 and H_α2, respectively.

We have

x_α1+α2(u) =

Exercise 11 Check (58) for various α^∨ and β.

To get started on the exercise, suppose α^∨ =α₁^∨ and β =α1 +α2. To calculate the inner producthα^∨, βi, we embed the root system in Euclidean space by the usual type C embedding (Example 3). In this embedding

α1 =e1 −e2 = (1,−1), α2 = 2e2 = (0,2).

Although we have been avoiding using (54) we use it now for the purpose of com-puting inner products, and find that

α^∨₁ = (1,−1), α^∨₂ = (0,1).

Therefore β = α₁ +α₂ = (1,1) and hα^∨₁, βi = 0. Indeed, h_α^∨

1(t) and x_α1+α2(u) commute, confirming (58).