Seminar Sophus Lie 2 (1992) 153–157

Weyl groups of disconnected Lie groups

Karl-Hermann Neeb

Introduction

Let G be a real Lie group and 𝔤 its Lie algebra. If one studies the exponential function exp : 𝔤 → G and fibrations of the set of regular elements of G over the set of all Cartan subalgebras of 𝔤, then one needs a general concept of a Cartan subgroup and a Weyl group. So let 𝔥 ⊆ 𝔤 be a Cartan subalgebra, i.e., a nilpotent subalgebra which is self-normalizing. Then we consider the adjoint action of the complexification 𝔥_{C} on the complexified Lie algebra 𝔤_{C} given by ad : 𝔥_{C} → der 𝔤_{C}, ad(X)(Y) = [X, Y].

For α ∈ 𝔥C^{∗} we set

𝔤C^{α} := ⋂_{X∈𝔥C} ⋃_{n∈IN} ker ( ad X − α(X) id )^{n},

and call it the α-root space. We write

Δ := Δ(𝔤_{C}, 𝔥_{C}) := {α ∈ 𝔥C^{∗} \ {0}: 𝔤C^{α} ≠ {0}}

for the set of roots.

Now let N(𝔥) := {g ∈ G: Ad(g)𝔥 = 𝔥} be the normalizer of 𝔥 in G. Note that for g ∈ N(𝔥) and γ := Ad(g)|_{𝔥C} we have that

Ad(g)𝔤C^{α} = 𝔤C^{γ.α},

where γ.α = α ∘ γ^{−1} . So N(𝔥) acts on the finite set Δ such that the effectivity kernel of this action is given by

C(𝔥) := {g ∈ N(𝔥): (∀α ∈ Δ)α ∘ Ad(g)|_{𝔥} = α}.

This group is called the Cartan subgroup associated to 𝔥. We note in passing that this definition of a Cartan subgroup is consistent with the standard definitions for semisimple Lie groups or algebraic groups (cf. [3]).

We define the Weyl group as

W := W(G, 𝔥) := N(𝔥)/C(𝔥).

This group is naturally isomorphic to the image of N(𝔥) in the group of permutations of Δ. It follows in particular that it is finite.

So it is very natural to ask which finite groups may occur as Weyl groups. In this note we prove the following theorem:

Theorem 1. Let Γ be a finite group. Then there exists a real Lie group G and a Cartan subalgebra 𝔥 ⊆ 𝔤 such that

W(G, 𝔥) ≅ Γ.

This contrasts the following result for connected Lie groups (cf. [3, V.12]) which shows that the Weyl groups of connected Lie groups are Weyl groups of semisimple groups.

Theorem 2. Let G be a connected Lie group and 𝔥 ⊆ 𝔤 a Cartan algebra. Then there exists a Levi decomposition 𝔤 ≅ 𝔯× 𝔰 such that 𝔥 ∩ 𝔰 is a Cartan subalgebra of 𝔰 and

W(G, 𝔥) ≅ W(S, 𝔥 ∩ 𝔰),

where S = ⟨exp 𝔰⟩ is the analytic subgroup corresponding to 𝔰.

Realization of finite groups as Weyl groups

In this section we prove Theorem 1. First we strip off the Lie algebra context to see what we have to prove about finite groups.

Let 𝔥 be an abelian real Lie algebra, Δ ⊆ 𝔥^{∗} \ {0} a finite generating subset, and

Γ := Aut(Δ) := {γ ∈ Gl(𝔥^{∗}) : γ.Δ = Δ}.

We define a representation of 𝔥 on V := IR^{Δ} by

X.v_{α} = α(X)v_{α} ∀X ∈ 𝔥, α ∈ Δ,

where v_{α}(β) = δ_{α,β}. In addition, we define an action of Γ on V by

γ.v_{α} = v_{γ.α} ∀γ ∈ Γ, α ∈ Δ.
Let X ∈ 𝔥, α ∈ Δ, and γ ∈ Γ. Then
γ.(X.v_{α}) = α(X)v_{γ.α}

and

(∗)  X.(γ.v_{α}) = (γ.α)(X)v_{γ.α} = α(γ.X)v_{γ.α},

where Γ acts on 𝔥 ≅ (𝔥^{∗})^{∗} via

γ.X := (γ^{∗})^{−1}.X.

Let H be a simply connected group with L(H) ≅ 𝔥. Then the representation of 𝔥 on V integrates to a representation of H on V with

exp(X).v_{α} = e^{α(X)}v_{α} ∀X ∈ 𝔥, α ∈ Δ.

Now (∗) shows that the actions of H and the action of Γ on H ≅ 𝔥 define an action of the semidirect product H × Γ. So we have a representation of the group H × Γ on V and we can form the Lie group

G := V × (H × Γ) ≅ (V × H)× Γ.

Lemma 2. The Lie group G has the following properties:

(i) 𝔥 is a Cartan algebra of 𝔤 := L(G).

(ii) Z(G) = {1} and Ad(G) ≅ G.

(iii) N(𝔥) = Ad(H × Γ).

(iv) Δ(𝔤_{C}, 𝔥_{C}) = Δ.

(v) C(𝔥) = Ad(H).

(vi) W(G, 𝔥) ≅ Γ.

Proof. (i) Since 𝔥 is abelian, it remains to show that 𝔥 is self-normalizing in 𝔤 ≅ V × 𝔥. This follows immediately from the assumption that 0 ∉ Δ which entails that

V_{fix} := {v ∈ V : 𝔥.v = {0}} = {0}.

(ii) Let g = (v, h, γ) and α ∈ Δ. Then

g(v_{α}, 1, 1)g^{−1} = (hγ.v_{α}, 1, 1) = (e^{γ.α(log h)}.v_{γ.α}, 1, 1).

If Ad(g) = 1, then this calculation shows that γ = 1 and α(log h) = 0 for all α ∈ Δ. Since Δ spans 𝔥^{∗}, it follows that log h = 0, i.e., h = 1. So g = (v, 1, 1) and since H acts on V without non-zero fixed points, it follows that v = 0. This proves that ker Ad = {1} and in particular that Z(G) = {1}. (iii) It is clear that H × Γ normalizes 𝔥. The other implication follows from (i). (iv) This is immediate from (i) and the construction of 𝔤.

(v) Using (iv) and (∗), we see that the action of H × Γ on Δ(𝔤_{C}, 𝔥_{C}) corresponds to the action of Γ on Δ. Hence the pointwise stabilizer coincides with H. (vi) This is a consequence of (iii) and (v).

Given a real vector space 𝔥 and a generating subset Δ ⊆ 𝔥^{∗} \ {0}, Lemma 2 provides a Lie group G such that 𝔤 contains a Cartan subalgebra 𝔥 with W(G, 𝔥) ≅ Γ.

So we have the following problem: Given a finite group Γ. Find a vector space 𝔥 and Δ ⊆ 𝔥^{∗} \ {0} such that

Γ ≅ Aut(Δ) = {γ ∈ Gl(V) : γ(Δ) = Δ}.

Thus the proof of Theorem 1 is completed by the following proposition.

Proposition 3. Let G be a finite group and A := IR[G] its group algebra. Then there exists a finite subset Δ ⊆ A^{∗} such that

Aut(Δ) ≅ G,

where G acts on A by left translations. Proof. We identify IR[G] with IR^{G}. Then we have a basis given by the functions ε_{g}(g′) = δ_{gg′}. We also consider A as a Hilbert space with respect to normalized Haar measure, i.e., {ε_{g}: g ∈ G} is an orthonormal basis. Using this Hilbert space structure, we identify A with its dual A^{∗}.

We choose an injective function m: G → IN with m(g) > 2 for all g ∈ G and set n(g, g′) := m(g^{−1}g′). We define the finite set

Δ := {ε_{g} : g ∈ G} ∪ {ε_{g} + n(g, g′)ε_{g′} : g, g′ ∈ G}.

Let C := ∑_{g∈G} IR^{+}ε_{g} denote the cone of positive functions in A = IR^{G}. This is a simplicial cone in the real vector space A. Let γ ∈ Gl(A) with γ(Δ) = Δ. Then γ(C) = C is a consequence of the fact that C is the smallest cone containing Δ. Thus γ preserves the set {IR^{+}ε_{g} : g ∈ G} of one-dimensional faces of C. It follows that for each g ∈ G the element γ(ε_{g}) is contained in a ray IR^{+}ε_{g′} for some g′ ∈ G. On the other hand IR^{+}ε_{g′} ∩Δ = ε_{g′} so that γ(ε_{g}) = ε_{g′}. Thus there exists a bijective mapping

η : G → G with γ(ε_{g}) = ε_{η.g}.

We claim that γ is left multiplication by η.1. To see this, let g, g′ ∈ G and f_{g,g′} := ε_{g} + n(g, g′)ε_{g′}. Then γ(f_{g,g′}) is an element in the two-dimensional face spanned by ε_{η.g} and ε_{η.g′} which is contained in Δ, so

γ.f_{g,g′} ∈ {f_{η.g,η.g′}, f_{η.g′,η.g}}.

Since

γ.f_{g,g′} = ε_{η.g} + n(g, g′)ε_{η.g′},

the equality

γ.f_{g,g′} = f_{η.g′,η.g} = ε_{η.g′} + n(η.g′, η.g)ε_{η.g}

is excluded by the fact that the coefficient n(η.g′, η.g) of ε_{η.g} is greater than 1. Whence

ε_{η.g} + n(g, g′)ε_{η.g′} = γ.f_{g,g′} = f_{η.g,η.g′} = ε_{η.g} + n(η.g, η.g′)ε_{η.g′}.

This leads to

m(g^{−1}g′) = n(g, g′) = n(η.g, η.g′) = m((η.g)^{−1}η.g′)

and, by the injectivity of m,

g^{−1}g′ = (η.g)^{−1}η.g′,

i.e., g(η.g)^{−1} = g′(η.g′)^{−1} . With g′ = 1 this identity yields g(η.g)^{−1} = (η.1)^{−1} , or, equivalently,

η.g = (η.1)g ∀g ∈ G.

This proves that Aut(Δ) consists of left multiplications in IR[G].

If, conversely, g ∈ G, then the left multiplication λ_{g}: x ↦ gx preserves Λ because it preserves the set {ε_{g}: g ∈ G} and

n(λ_{g}.x, λ_{g}.y) = m(x^{−1}g^{−1}gy) = m(x^{−1}y) = n(x, y)

shows that it also preserves the set {f_{g,g′} : g, g′ ∈ G}.

References

[1] Hofmann, K. H., A memo on the exponential function and regular points, Arch. Math. 59(1992), 24–37.

[2] —, Near-Cartan algebras and groups, Seminar Sophus Lie 2 1992, 135– 151.

[3] Neeb, K.– H., A memo on Cartan subgroups and regular elements, Seminar report, Technische Hochschule Darmstadt, 1992.

Fachbereich Mathematik Technische Hochschule Darmstadt Schloßgartenstraße 7 D-6100 Darmstadt, Germany neeb@mathematik.th-darmstadt.de

Received November 4, 1992