Strong real forms and the Kac classification

Jeffrey Adams

October 21, 2005

This paper is expository. It is a mild generalization of the Kac classification of real forms of a simple Lie group to strong real forms. The basic reference for strong real forms in this language is [2]. For the Kac classification we follow [6]. There is also a treatment in [4], in slightly different terms.

1 Real forms and strong real forms

Let G be a reductive algebraic group. We will occasionally identify algebraic groups with their complex points. We have the standard exact sequence

(1.1)  1 → Int(G) → Aut(G) → Out(G) → 1

where Int(G) ≃ G/Z(G) is the group of inner automorphisms of G, Aut(G) is the automorphims of G, and Out(G) = Aut(G)/Int(G).

Definition 1.2 1. A real form of G is an equivalence class of involutions in Aut(G), where equivalence is by conjugation by G, i.e. the action of Int(G).

2. A traditional real form of G is an equivalence class of involutions, where equivalence is by the action of Aut(G).

The real form defined by θ has a maximal compact subgroup whose complexification is K = G^{θ}.

Remark 1.3 A real forms are also defined by an antiholomorphic involution σ of G(ℂ), i.e. G(ℝ) = G(ℂ)^{σ}. Given θ choose an antiholomorphic involution σ_{0} so that θσ_{0} = σ_{0}θ and G(ℝ)_{0}^{σ} is compact. Then the real form defined by θ is given by σ = θσ_{0}. See [?, Section VI.2].

We say two involutions θ, θ′ ∈ Aut(G) are inner to each other, or in the same inner class, if they have the same image in Out(G). Such a class is determined by an involution γ ∈ Out(G), and we refer this inner class as the real forms of (G, γ).

We will work entirely in a fixed inner class, so fix an involution γ ∈ Out(G).

Fix a splitting datum for the exact sequence (1.1). This is a set (H, B, {X_{α}}) consisting of a Cartan subgroup H, a Borel subgroup B containing H, and a set of simple root vectors. This induces a splitting Out(G) → Aut(G) of (1.1), and we let θ be the image of γ in Aut(G). Thus θ is an involution of G, it corresponds to the “most compact” real form in the given inner class. We let K = G^{θ}.

Remark 1.4 Suppose G is simple and simply connected. It does not necessarily follow that K is simply connected; it is not simply connected if and only if the real form G = G(ℝ) of G corresponding to K has a non-linear cover. Since θ is the most compact inner form of (G, γ) K has a chance to be simply connected. In fact this holds unless G = SL(2n + 1), in which case K = SO(2n + 1) and π_{1}(K) = ℤ/2ℤ. This exception is due to the fact that Δ_{θ} (cf. Lemma B.1) is not reduced in this case. See the table in Section 3.1.

For most of these notes G will be semisimple, or even simple. Let Δ = Δ(G, H) be the root system of H in G, and let D = D(Δ) be the Dynkin diagram of Δ. A choice of splitting datum induces an isomorphism

(1.5)  Out(𝔤) ≃ Aut(D)

Furthermore Out(G) ⊂ Out(𝔤), with equality if G is simply connected or adjoint. Thus γ is given by an involution of D.

Let

G^{Γ} = G ⋊ ⟨δ⟩

where δ^{2} = 1 and δgδ^{−1} = θ(g).

Definition 1.6 A strong real form of (G, γ) is an equivalence class of elements x ∈ G^{Γ}, satisfying x ∉ G, and x^{2} ∈ Z(G), where equivalence is by conjugation by G.

The map x → θ_{x} = int(x) is a surjection from the strong real forms of (G, γ) to the real forms of (G, γ). Let

H^{Γ} = H ⋊ ⟨δ⟩ ⊂ G^{Γ}.

Let T be the identity component of H^{θ}, and A be the identity component of H^{−θ} = {h ∈ H | θ(h) = h^{−1}}. Then H = T A. Let

T^{Γ} = T × ⟨δ⟩ ⊂ H^{Γ}.

Remark 1.7 We may write

(1.8)  H ≃ ℂ^{∗a} × ℂ^{∗b} × (ℂ^{∗} × ℂ^{∗})^{c}

where θ acts trivially on the first a factors, by inverse on the next b, and θ(z, w) = (w, z) on each of the last c terms. Note that if b ≠ 0 then T is a proper subset of H^{θ}. This happens, for example, in SO(3, 1).

A key observation is that every element of Hδ is conjugate to an element of T δ, since A acts by conjugation on Hδ by multiplication by A. That is for a ∈ A, h ∈ H, a(hδ)a^{−1} = ah(δa^{−1}δ)δ = ahθ(a^{−1})δ = a^{2}hδ. Since A is connected every element has a square root, so this gives multiplication by an arbitary element of A. Therefore taδ is conjugate to tδ ∈ T δ. (In fact we could replace T with H/A = T/T ∩ A, see ?.)

Lemma 1.9 Suppose x ∈ Gδ is a semi-simple element (i.e. x = gδ with g ∈ G semisimple). Then x is G-conjugate to an element of T δ.

Proof. Write x = gδ and choose a Cartan subgroup H′ containing g. Write H′ = T′A′ as usual and g = ta accordingly. As above we may assume a = 1. Since T is a Cartan subgroup of K we may choose k ∈ K so that ktk^{−1} ∈ T, and then k(tδ)k^{−1} ∈ T δ.

Let W = Norm_{G}(H)/H. Then θ acts on W, and we let W^{θ} be its fixed points. Note that W^{θ} acts naturally on T, A and T ∩ A.

Lemma 1.10

(1.11)  W (K^{0}, T) ≃ W (G, H)^{θ}

Remark 1.12 In almost all cases K is connected, and T is a Cartan subgroup of K. if K is not connected hen H^{θ} is a Cartan subgroup of K. In this case H^{θ} = T Z(G), and W (K, H^{θ}) ≃ W (K^{0}, T). This is the case, for example, if G = SO(2n) and K = S[O(2n − 1) × O(1)] ≃ O(2n − 1).

Remark 1.13 One consequence of Lemma 1.10 is this: if w ∈ W^{θ} we may choose a representative g ∈ Norm_{G}(H) of w to be in K.

Lemma 1.14 Suppose x, x′ ∈ T δ are G-conjugate. Then there exists g ∈ Norm_{G}(T δ) so that gxg^{−1} = x′.

Thus G-conjugacy of elements of T δ is controlled by the group W_{δ} of the next definition.

Definition 1.15

(1.16)  W_{δ} = Norm_{G}(T δ)/Cent_{G}(T δ)

It is well known that Cent_{G}(T) = H. Therefore Norm_{K^{0}}(T) ⊂ Norm_{G}(H) and we obtain a map

W (K^{0}, T) ↪ W (G, T)

whose image is contained in W (G, T)^{θ}.

Proposition 1.17

(1.18)  W_{δ} ≃ W^{θ} ⋉ (A ∩ T).

The subgroup W^{θ} is the stabilizer of δ in W_{δ}, and acts on T via its natural action. The subgroup A ∩ T acts on T δ by multiplication.

Proof. It is well known that Cent_{G}(T) = H (every root α ∈ Δ(G, H) is non-trivial on T, since there are no real roots). Therefore Norm_{G}(T) = Norm_{G}(H). Thus

(1.19)(a)  Norm_{G}(T δ) = {g ∈ Norm_{G}(H) | gδg^{−1} ∈ T }.

It is also clear that

(1.19)(b)  Cent_{G}(T) = {g ∈ H | gδg^{−1} = δ} = H^{θ}

Therefore

(1.19)(c)  W_{δ} = {g ∈ Norm_{G}(H) | gδg^{−1} ∈ T }/H^{θ}.

We also have

(1.19)(d)  W^{θ} = {g ∈ Norm_{G}(H) | gδg^{−1} ∈ H}/H

If gδg^{−1} = ta ∈ H, choose b ∈ H so that b^{2} = a. Then (bg)δ(bg)^{−1} = t ∈ T. It follows that the natural map W_{δ} → W^{θ} is a surjection. The kernel is

(1.20)  {h = ta ∈ H | a^{2} ∈ T }/H^{θ}

Let A_{0} = {a ∈ A | a^{2} ∈ T }, so the kernel is

(1.21)  T A_{0}/H^{θ} = T A_{0}/T A^{θ} = A_{0}/A^{θ}.

Now the map a → a^{2} takes A_{0} onto A ∩ T and there is an exact sequence

(1.22)  1 → A^{θ} → A_{0} → A ∩ T → 1

Therefore A_{0}/A^{θ} ≃ A ∩ T. See Remark 1.24.

Putting this together we have an exact sequence

(1.23)  1 → A ∩ T → W_{δ} → W_{θ} → 1

Define a splitting of (1.23) by taking w ∈ W^{θ} to the unique preimage in W_{δ} fixing δ. This exists by Lemma 1.10: given w ∈ W^{θ} there exists g ∈ Norm_{K}(H) ⊂ Norm_{G}(H) representing w. It is easy to see this is a well defined splitting.

The action of W^{θ} on T δ is clear. For a ∈ A ∩ T choose b ∈ A_{0} so that b^{2} = a. Then b(tδ)b^{−1} = btθ(b)^{−1}δ = b^{2}tδ = a(tδ), so A ∩ T acts by multiplication.

Remark 1.24 With respect to the decomposition (1.8) we have

\begin{aligned}A_{0} & ≃ (ℤ/2ℤ)^{b} × (ℤ/4ℤ)^{c} \\ A^{θ} & ≃ (ℤ/2ℤ)^{b} × (ℤ/2ℤ)^{c} \\ A ∩ T & ≃ 1 × (ℤ/2ℤ)^{c} \end{aligned}

where ℤ/4ℤ = {±(1, 1), ±(i, −i)} ⊂ ℂ^{∗} × ℂ^{∗}. This makes (1.22) explicit.

Proposition 1.25 The strong real forms of (G, γ) are are parametrized by elements x of T δ satisfying x^{2} ∈ Z, modulo the action of W_{δ}.

It is convenient to mod out by the translations in T ∩ A; this amounts to replacing T with H/A ≃ T/T ∩ A. Let

(1.26)  \overline{T} = T/T ∩ A, \overline{T}^{Γ} = \overline{T} × ⟨δ⟩

Note that W^{θ} acts on \overline{T}. Also every element of T ∩ A has order 2, so the condition x^{2} ∈ Z for x ∈ \overline{T} is well defined. This gives:

Proposition 1.27 The strong real forms of (G, γ) are are parametrized by elements x of \overline{T} δ satisfying x^{2} ∈ Z, modulo the action of W^{θ}.

One advantage of \overline{T} δ over T δ is that Z acts naturally on \overline{T}, via the isomorphism \overline{T} ≃ H/A.

To compute the orbits of W_{δ} on \overline{T} δ we pass to the tangent space, in which W_{δ} becomes an affine Weyl group. See the Appendix for some generalities about affine root systems and Weyl groups.

2 Affine Weyl group and strong real forms

We are interested in computing the orbits of W^{θ} acting on \overline{T} δ (Proposition 1.25).

Let π : E → \overline{T} δ be the tangent space of \overline{T} δ at δ. We recall a few definitions from the Appendix. The space E is an affine space, with group of translations 𝔱 = Lie(T). The space of affine linear functions E → E is denoted Aff(E, E).

Definition 2.1 Suppose B is a subgroup of Aut(\overline{T} δ). Let B̃ be the lift of B to Aff(E, E). That is

B̃ = {ϕ ∈ Aff(E, E) | ϕ factors to an element of B}.

From Proposition 1.27 we see:

Lemma 2.2 Strong real forms of G are parametrized by elements X of E satisying π(X)^{2} ∈ Z modulo the action of W̃^{θ}.

We consider the problem of finding a fundamental domain for the action of W̃^{θ} on E, and return later to the question of finding the subset of X such that π(X)^{2} ∈ Z.

We first suppose G is simply connected. From the Appendix (Definitions B.7 and B.9 and Proposition B.12)

W̃^{θ} = W_{aff} ≃ W^{θ} ⋉ L_{sc}

(the last isomorphism depending on a choice of δ̃ lying over δ). Also W_{aff} is the affine Weyl group of the affine root system D_{Aff}. The underlying finite root system is Δ_{θ}.

There is a standard choice of a fundamental domain for the action of W_{aff} on E. Choose a set of simple roots α̃_{0}, …, α̃_{n} of Δ_{aff}, and let

\overline{D} = \{e ∈ E | α̃_{i}(e) ≥ 0, i = 0, …, n\}.

If we choose δ̃ then we may identify E with V, and write α̃_{i} = (α_{i}, 0) (i = 1, …, n) and α̃_{0} = (α_{0}, c). Let β = −α_{0}; recall β is the highest long (resp. short) root of Δ if c = 1 (respectively c = 2). Then

\overline{D} = \{v ∈ V | α_{i}(v) ≥ 0 (i = 1, …, n), β(v) ≤ c\}.

If G is not simply connected then W_{aff} ⊂ W̃^{θ}, and W̃^{θ} is an extended affine Weyl group. Its fundamental domain will be a quotient of D by a finite group.

Definition 2.3 Let

(2.4)  L(G) = X_{∗}(T/T ∩ A).

In particular we have

(2.5)  L(G)/X_{∗}(T) ≃ T ∩ A

Lemma 2.6

L(G) = ⟨\frac{1}{c} ∑_{k=0}^{c−1} θ^{k}(γ^{∨}) | γ ∈ X_{∗}(H)⟩

If c = 1, 2 we have

(2.7)  L = \{\frac{1}{2}(α^{∨} + θα^{∨}) | α ∈ X_{∗}(H\} (c = 1, 2).

If G is simply connected then L(G) = L_{sc} (Definition B.9). Lemma 2.8 Setting L = L(G) we have an exact sequence

(2.9)(a)  1 → L → W̃^{θ} → W^{θ} → 1

Given δ̃ we obtain a splitting of (2.9)(a), so

(2.9)(b) W̃^{θ} ≃ W^{θ} ⋉ L.

If G is simply connected then (2.9)(a-b) reduce to (B.13)(a-b).

To find a fundamental domain for W̃^{θ} we relate it to W_{aff}. Lemma 2.10 We have an exact sequence

(2.11)  1 → W_{aff} → W̃^{θ} → L/L_{sc} → 1

Given δ̃ we obtain a splitting taking L/L_{sc} to the stabilizer of D. Thus

(2.12) W̃^{θ} ≃ W_{aff} ⋊ L/L_{sc}

and L/L_{sc} acts as automorphisms of D.

Recall we are given (Δ, θ), to which we have associated the affine root system Δ_{aff}, with Dynkin diagram D_{Aff}. See the Appendix. Lemma 2.13 The stabilizer of D in the Euclidean group of E is isomorphic to the automorphism group of D_{Aff}.

Thus we have an action of L/L_{sc} on D_{Aff}. It behooves us to understand L/L_{sc}.

2.1 The group L/L_{sc}

From (B.11) we have

L/L_{sc} = \frac{⟨\{\frac{1}{2}(γ^{∨} + θγ^{∨}) | γ^{∨} ∈ X_{∗}(H)\}⟩}{⟨\{\frac{1}{2}(α^{∨} + θα^{∨}) | γ^{∨} ∈ R^{∨}\}⟩}

Let G_{sc} be the simply connected cover of G, with center Z_{sc} = Z(G_{sc}). We have an exact sequence

1 → π_{1} → G_{sc} → G → 1

with π_{1} = π_{1}(G) ⊂ Z_{sc}. Write H_{sc} = T_{sc}A_{sc} for the Cartan subgroup in G_{sc} with image H.

Lemma 2.14

(2.15)  L/L_{sc} ≃ π_{1}/π_{1} ∩ A_{sc}

Proof. A standard fact is that π_{1} ≃ X_{∗}(H)/R^{∨}. The map γ^{∨} → \frac{1}{2}(γ^{∨} +θγ^{∨}) takes X_{∗}(H) onto L and factors to a surjection

π_{1} ↠ L/L_{sc}.

The kernel is

{γ^{∨} ∈ X_{∗}(H) | (1 + θ)γ^{∨} ∈ (1 + θ)R^{∨}}/R^{∨}

If (1+θ)γ^{∨} = (1+θ)μ^{∨} for some μ^{∨} ∈ R^{∨} then (1+θ)(γ^{∨}−μ^{∨}) = 0. So we may replace the numerator with {γ^{∨} | (1+θ)γ^{∨} = 0}. This says exp(2πiγ^{∨}) ∈ A_{sc}, so the kernel is π_{1} ∩ A_{sc}.

Remark 2.16 Note that

(1 − θ)π_{1} ⊂ π_{1} ∩ A_{sc} ⊂ π_{1}^{−θ}

and both inclusions may be proper. If G is adjoint then π_{1} = Z_{sc} and one can see Z_{sc} ∩ A_{sc} = (1 − θ)Z_{sc}, which gives

(2.17)  L_{ad}/L_{sc} = Z_{sc}/(1 − θ)Z_{sc}.

However it is not easy to describe π_{1} ∩ A_{sc} in general.

Definition 2.18 Let

(2.19)  π_{1}^{†} = π_{1}/π_{1} ∩ A_{sc}

Let τ : π_{1}^{†} → Aut(D_{Aff}) be the action of π_{1}^{†} on the affine Dynkin diagram via Lemmas 2.10, 2.13 and (2.15).

Here is another description of τ. First take G to be simply connected, so Z = Z_{sc}. Note that Z acts by left multiplication on Hδ and therefore on \overline{T} δ. Explicitly z = ta ∈ Z acts on \overline{T} δ by multiplication by t. Although t, a are only defined up to T ∩ A, this action is well defined on \overline{T} δ. Clearly this action factors to Z/Z ∩ A, lifts to an action on E, and induces actions of Z/Z ∩ A on D and D_{Aff}.

Suppose z = ta = exp(2πiγ^{∨}) with γ^{∨} ∈ P^{∨}. Then t = exp(2πi\frac{1}{2}(γ^{∨} + θγ^{∨})), and it follows that under the isomorphism (2.15) L_{ad}/L_{sc} acts by translation on E.

Now drop the assumption that G is simply connected. Then π_{1}(G) ⊂ Z_{sc} acts on D and D_{Aff} by the preceding construction, and this action factors to an action of π_{1}^{†}(G).

Lemma 2.20 We may parametrize \overline{D} as \{(a_{0}, …, a_{n})\} where a_{i} ≥ 0 and

(2.21)  ∑_{i=0}^{n} n_{i}a_{i} = \frac{1}{c}.

Here (a_{0}, …, a_{n}) corresponds to the element X of D satisfying

α_{i}(X) = a_{i} (i = 1, …, n)

Lemma 2.22 Suppose (a_{0}, a_{1}, …, a_{n}) satisfies (2.21), and let X ∈ D be the corresponding element. Then x = π(X) ∈ \overline{T} δ satisfies x^{m} ∈ Z if and only if ma_{i} ∈ ℤ for all i = 0, …, n.

Example 2.23 Take m = 1. We must take c = 1 and each a_{i} = 0 or 1. We conclude from (2.21) that Z is in bijection with the nodes of D̃ with label 1.

Given m choose integers b_{i} and let a_{i} = b_{i}/m (0 ≤ i ≤ n). Then (a_{0}, …, a_{n}) corresponds to an element of D if

(2.24)  c ∑_{i=0}^{n} n_{i}b_{i} = m

To complete our classification of strong real forms we take m = 1 or 2.

Definition 2.25 A Kac diagram for (G, γ) is a subset S of D_{Aff} satisfying c ∑_{i∈S} n_{i} ≤ 2.

Clearly |S| ≤ 2 and n_{i} ≤ 2 for all i ∈ S.

Theorem 2.26 Fix G and an inner class γ of real forms. Let c = order(γ). Let θ be the fundamental real form in the given inner class. Let Δ be the root system of G, Δ_{θ} the quotient of Δ by θ, and D_{Aff} the affine Dynkin diagram associated to Δ_{θ}.

The strong real forms of (G, γ) are parametrized by Kac diagrams for D_{Aff}, modulo the action of π_{1}^{†}(G) on D_{Aff}.

Suppose S is a Kac diagram corresponding to a real form, with (complexified) maximal compact subgroup K_{S}. Then the Dynkin diagram of K_{s} is obtained by by deleting the nodes of S from D_{Aff}.

For the usual classification of real forms see the next section.

For example, a compact group is given by m = 1, c = 1 and S = {i} with n_{i} = 1.

Suppose m = 2. If c = 1, then S = {i} with n_{i} = 2, or S = {i, j} with n_{i} = n_{j} = 1. If c = 2 then S = {i} with n_{i} = 1.

2.2 The Kac classification of real forms

The Kac classification of real forms of 𝔤 amounts to taking G to be the adjoint group. In this case π_{1}^{†}(G) = Z_{sc}/(1 − θ)Z_{sc} (2.17). Recall (2.19) acts by τ on D_{Aff} (Definition 2.18).

Theorem 2.27 Traditional real forms of (𝔤, γ) are parametrized by subsets S as in Theorem 2.26, modulo the action of Aut(D_{Aff}).

Real forms of (𝔤, γ) are parametrized by subsets S modulo the action of Z_{sc}/(1 − θ)Z_{sc}.

Proof. The second statement is an immediate consequence of Theorem 2.26. The first follows from the following Lemma.

Remark 2.28 This also gives the classification for G either simply connected or adjoint. For general G equivalence will be by the subgroup stabilizing Z(G).

Lemma 2.29 We have a split exact sequence

1 → π_{1}^{†}(G) → Aut(D) → Out(G) → 1

or equivalently

1 → π_{1}^{†}(G) → Aut(D_{Aff}) → Aut(D_{θ}) → 1

Here D_{θ} is the Dynkin diagram of Δ_{θ}, the underlying finite root system of D_{Aff}. See the Appendix.

Remark 2.30 If θ = 1 and G is simply connected this becomes

1 → Z → Aut(D_{Aff}) → Aut(D) → 1

If θ ≠ 1 then Aut(D_{θ}) = 1 and we have

π_{1}^{†} ≃ Aut(D_{Aff})

See [6, Exercise 15, page 217]. For an explicit formula for the map Z → Aut(D_{Aff}) see [3, Chapter VI, §2.3, Proposition 6].

3 Simplified Kac Diagrams and Vogan Diagrams

If γ ≠ 1 the classification of real forms via the Kac diagram is quite subtle, due to its use of the extended Dynkin diagram of Δ_{θ}, rather than that of Δ. Here is a version using the extended Dynkin diagram of Δ.

So fix (G, γ) with G simple and γ ≠ 1. Choosing a splitting datum, in particular a Cartan subgroup H we obtain the fundamental automorphism θ of G as in Section 1. Write H = T A as usual.

For simplicity we assume G is adjoint, so strong real forms and real forms coincide. Suppose γ ≠ 1. By Proposition 1.27 the real forms of (G, γ) are parametrized by elements t ∈ T of order 2 (corresponding to x = tδ ∈ \overline{T} δ), modulo T ∩ A and conjugation by W^{θ}.

On the other hand the real forms of (G, 1) are parametrized by elements of H of order 2, modulo conjugation by G. If two elements of t are conjugate by W then they are necessarily conjugate by W^{θ}. If S is the Kac diagram of a real form of (G, 1), then the corresponding element h is in T if and only if S is θ-invariant. This gives a surjective map from

θ − invariant Kac diagrams for (G, 1) → strong real forms of (G, γ)

This map is not injective: on the left hand side equivalence is by the action of W^{θ}, and on the right by W^{θ} and A ∩ T. It turns out that if we require that S is pointwise fixed by θ then we get a bijection.

Proposition 3.1 Given (G, γ) let D_{Aff} be the extended Dynkin diagram of Δ = Δ(G, H). Then real forms of (G, γ) are parametrized by Kac diagrams S for which each node of S is fixed by θ, modulo Aut(D_{Aff}). That is, S is a set of θ-fixed nodes of D_{Aff}, such that c ∑_{i∈S} n_{i} ≤ 2.

To be honest there is some case-by-case checking here. One point is this. Suppose α is a complex root, and n_{α} = 1. Then S = {α, θα} defines an element t of T of order 2, and a real form of (G, 1). It also defines a real form of (G, γ), but this one is obtained from another set S which is pointwise fixed.

3.1 Vogan Diagrams

We continue to assume (G, γ) and H have been fixed, and θ is the fundamental real form of G. Let D be the Dykin diagram of G. Suppose θ′ is a real form of (G, γ) and B is a θ-stable Borel subgroup of G containing H. Associated to this data is a Vogan Diagram: color each of the θ-fixed nodes of D black if the cooresponding imaginary root is non-compact, and white otherwise. See [?, Section VI.8]. Alternatively, let S be the subset (of black nodes) of the θ-fixed nodes of D. This gives a map from real forms of (G, γ) to Vogan diagrams. This map is not injective: it depends on the choice of B. If we choose B to be the “Borel de Siebenthal” choice [?, Theorem 6.96], i.e. for which at most one simple root is non-compact, then we get a set S with at most one element.

The is closely related to the simplified Kac diagram. Here is the precise statement.

Proposition 3.2 Suppose S is a modified Kac diagram of a real form. If S contains a node with label 1 we may assume (via the action of Z_{sc}) this is the affine node. Deleting this node we obtain a subset of the finite Dynkin diagram. This is the Vogan diagram of the real form.

Conversely suppose S is a Vogan diagram with at most one node, corresponding to a real form of G. Also assume it satisfies the condition in the last line of [?, Thoerem6.96]; equivalently the label on this node is ≤ 2. If S is empty this is the compact form. Suppose S = {i}. The Kac diagram of this real form is S ∪ {0} if n_{i} = 1, and S if n_{i} = 1.

Remark 3.3 One of the subtleties of the Vogan diagram is that we do not need a diagram S = {i} if n_{i} ≥ 3. The fact that such Kac diagram is not needed is explained by (2.24).

Appendix: Affine root systems and Weyl groups

Let V be a real vector space of dimension n and E an affine space with translations V. That is V acts simply transitively on E, written v, e → v +e. A function If E, E′ are affine spaces a function f : E → E′ is said to be affine if there exists a linear function df : V → V′ such that

(A.1)  f(v + e) = df(v) + f(e) for all v ∈ V, e ∈ E.

In particular if E′ is one dimensional we say f is an affine linear functional. In this case df : V → ℝ, i.e. df ∈ V^{∗}. We say df is the differential of f. The set Aff(E) of all affine linear functionals is a vector space of dimension n + 1. The map f → df is a linear map from Aff(E) to V^{∗}, and this gives an exact sequence

(A.2)  0 → ℝ → Aff(E) → V^{∗} → 0.

The first inclusion takes x ∈ ℝ to the constant function f_{x}(e) = x for all e ∈ E; this satisfies df = 0.

Choose an element e_{0} ∈ E. This gives an isomorphism V ≃ E via v → v + e_{0}. For λ ∈ V^{∗} let s(λ)(v + e_{0}) = λ(v). This defines a splitting of (A.2):

Lemma A.3 Given e_{0} we obtain an isomorphism

(A.4)(a)  Aff(E) ≃ V^{∗} ⊕ ℝ

According to this decomposition we write f ∈ Aff(E) as

(A.4)(b)  f = (λ, c).

We make the isomorphism (A.4)(a) explicit. In one direction f ∈ Aff(E) goes to λ = df and c = f(e_{0}). For the other direction (λ, c) goes to f ∈ Aff(E) defined by f(v + e_{0}) = λ(v) + c.

We now assume V is equipped with a positive definite non-degenerate symmetric form (,), and identify V and V^{∗}. In particular we may identify df with an element of V. Define (,) on Aff(V) by

(f, g) = (df, dg)

and for f ∈ Aff(E) not a constant function let

f^{∨} = \frac{2f}{(f, f)}.

The affine reflection s_{f} : V → V is

s_{f}(v) = v − f^{∨}(v)df

= v − f(v)(df)^{∨}

= v − \frac{2f(v)}{(f, f)}df

Definition A.5 (Macdonald [5]) An affine root system on E is a subset S of Aff(E) satisfying

1. S spans Aff(E), and the elements of S are non-constant functions, 2. s_{α}(β) ∈ S for all α, β ∈ S,

3. ⟨α^{∨}, β⟩ ∈ ℤ for all α, β ∈ S,

4. The Weyl group W = W (S) is the group generated by the reflections {s_{α} | α ∈ S}. We require that W acts properly on V.

The Weyl group W (S) is an affine Weyl group. The notions of simple roots Π(S) and Dynkin diagram D(S) are simlar to those for classical root systems. Also the dual S^{∨} of S defined in the obvious way is an affine root system, with Dynkin diagram D(S^{∨}) = D(S)^{∨}. Here the dual of a Dynkin diagram means the same diagram with arrows reversed, as usual.

Choose a base point e_{0} in E and write elements of Aff(E) as (λ, c) as in Lemma A.3.

Suppose Δ ⊂ V is a classical (not necessarily reduced) root system. If Δ is simply laced we say each root is long. Let Π = {α_{1}, …, α_{n}} be a set of simple roots. For each i let α̃_{i} = (α_{i}, 0), and let α̃_{0} = (−β, 1) where β is the highest root. Note that β is long. Then {α̃_{0}, …, α̃_{n}} is a set of simple roots of an affine root system denoted Δ̃.

Let D = D(Δ) be the Dynkin diagram of Δ. Let D̃ be the extended Dynkin diagram of D, i.e. obtained by adjoining −β where β is the highest root. Then the Dynkin diagram of Δ̃ is the extended Dynkin diagram of Δ, i.e.

D(Δ̃) = { \atop D(Δ)}.

We will use Δ (resp. S) to denote a typical classical (resp. affine) root system.

Suppose Δ is a classical root system with Dykin diagram D = D(Δ). Let and S = Δ̃, so D(S) = D̃. Then S^{∨} = (Δ̃)^{∨} is also an affine root system, with Dynkin diagram D(S^{∨}) = (D̃)^{∨}. If Δ is not simply laced then it is not necessarily the case that (Δ̃)^{∨} = { \atop (Δ^{∨})} or (D̃)^{∨} = { \atop (D^{∨})}. Note that D̃ is obtained from D by adding a long root, so (D̃)^{∨} has an extra short root. On the other hand { \atop (D^{∨})} is obtained from D^{∨} by adding an extra long root.

Theorem A.6 (Macdonald [5]) Every reduced, irreducible affine root system is equivalent to either Δ̃ or (Δ̃)^{∨} where Δ is a classical (not necessarily reduced) root system.

Remark A.7 A remarkable fact is that every reduced, irreducible affine root system is also obtained from a classical root system and involution, as discussed in the next section.

Affine root system and Weyl group associated to (Δ, θ)

Let Δ be an irreducible root system, and θ an automorphism of Δ preserving a set of simple roots. Thus θ corresponds to an automorphism of the Dynkin diagram D = D(Δ) of Δ. Let c ∈ {1, 2, 3} be the order of δ. Associated to (Δ, θ) is an affine root system, which we now describe.

The quotient Δ/θ is naturally a root system [7], which we denote Δ_{θ}. Here are the possibilities with θ ≠ 1. We list the finite root systems Δ, Δ_{θ}, the names of the affine root system according to [5] and [6], the simply connected group G with root system Δ, the real form of G corresponding to θ, and G^{θ}.

Δ  Δ_{θ}  Δ_{aff} Δ_{aff}  G  G(ℝ)  K
A_{2n}  BC_{n} { \atop BC_{n}} A_{2n}^{(2)} SL(2n + 1) SL(2n + 1, ℝ) SO(2n + 1)
A_{2n−1}  C_{n}  B̃_{n}^{∨} A_{2n−1}^{(2)} SL(2n)  SL(n, ℍ)  Sp(n)
D_{n}  B_{n}  C̃_{n}^{∨} D_{n}^{(2)} Spin(2n) Spin(2n − 1, 1) Spin(2n − 1)
E_{6}R  F_{4}  F̃_{4}^{∨} E_{6}^{(2)}  E_{6}  E_{6}(F_{4})  F_{4}
D_{4}, θ^{3} = 1 G_{2}  G̃_{2}^{∨} D_{4}^{(3)}  Spin(8)  G_{2}

As in section 1 there is an algebraic group G, and splitting data (H, B, {X_{α}}) so that Δ = Δ(G, H), and θ may be viewed as an automorphism of G preserving the splitting data. (For these purposes we may as well take G simply connected.) Then T = H^{θ} acts on 𝔤, and the set of roots Δ(G, T) ⊂ 𝔱^{∗} is a (possibly reduced) root system.

The following Lemma is more or less immediate.

Lemma B.1 Restriction from H to T defines isomorphisms

Δ(G, T) ≃ Δ_{θ}

and

W^{θ} ≃ W (Δ_{θ}).

Also Δ(K, T) is the reduced root system of Δ_{θ} (obtained by taking only the shorter of two roots α, 2α) and W (K, T) ≃ W (Δ_{θ}). See Remark 1.4.

Now T^{Γ} acts on the complex Lie algebra 𝔤 of G. Let Δ(G, T^{Γ}) be the set of roots, i.e. we have a root space decomposition

𝔤 = ∑_{α∈Δ(G,T^{Γ})} 𝔤_{α}.

Clearly restriction from T^{Γ} to T is a surjection Δ(G, T^{Γ}) → Δ(G, T).

If c = 1 this is simply Δ(G, T). For simplicity assume c = 2. Then Δ(G, T^{Γ}) may be thought of as a ℤ/2ℤ-graded root system. That is a character α of T^{Γ} is a pair (α_{0}, ϵ) with α_{0} ∈ Δ(G, T) ≃ Δ_{θ} and ϵ = ±1, where α_{0} = α|_{T} and ϵ = α(δ). We can define the reflection associated to α ∈ Δ(G, T^{Γ}) in the usual way, preserving Δ(G, T^{Γ}). To be precise, if α = (α_{0}, ϵ) and β = (β_{0}, δ) then

(B.2)  s_{α}(β) = (s_{α_{0}}(β_{0}), ϵδ(−1)^{⟨β,α^{∨}⟩}).

Let π : E → T δ be the universal cover. Then E is an affine space with translations 𝔱 = Lie(𝔱).

Suppose λ is a character of T^{Γ} → ℂ^{∗}. Note that λ is determined by its restriction to T δ. By the property of covering spaces λ lifts to a family of functions λ̃ : E → ℂ satisfying

λ(π(X)) = e^{2πiλ̃(X)}

i.e. dλ̃ = dλ, where the left hand side is in the sense of (A.1) and the right is the ordinary differential of λ. We say λ̃ lies over λ. Any two such functions differ by constant.

Definition B.3 The affine root system Δ_{aff} associated to (Δ, θ) is the set of affine functions in Aff(E) lying over Δ(G, T^{Γ}).

Note that the underlying finite root system, i.e. the differentials of all affine roots is Δ(G, T) ≃ Δ_{θ}, i.e.

d : Δ_{aff} ↠ Δ_{θ}

The following Lemma is an immediate consequence of the fact that Δ(G, T^{Γ}) is a root system in the sense of (B.2).

Lemma B.4 Δ_{aff} is an affine root system.

To be explicit, choose δ̃ ∈ E with π(δ̃) = δ. Suppose α ∈ T̂^{Γ}. To avoid excessive notation we write α for the differential of α restricted to T, rather than dα. Then in the decomposition of Lemma A.3 we may write the set of α̃ lying over α as

{(α, c) | e^{2πic} = α(δ)}

In particular note that the set of roots lying over α is

{(α, c) | c ∈ ℤ} if α(δ) = 1

or

\{(α, c) | c ∈ ℤ + \frac{1}{2}\} if α(δ) = −1

Similarly if δ has order 3 then c ∈ ℤ + \frac{1}{3} or ℤ + \frac{2}{3}.

For α ∈ Δ_{θ} let c_{α} = 1 if α is long, or \frac{1}{c} if α is short, where c = order(θ).

Proposition B.5 Let Δ_{aff} be the affine root system associated to (Δ, θ), and let c = order(θ) ∈ {1, 2, 3}. Then

Δ_{aff} = {(α, x) | x ∈ c_{α}ℤ}

Proposition B.6 Fix a set α_{1}, …, α_{n} of simple roots of Δ_{θ}. For each i let α̃_{i} = (α_{i}, 0). Let β be the highest (long) root of Δ = Δ_{θ} if c = 1 or the highest short root otherwise. Let

α̃_{0} = (−β, \frac{1}{c}).

Then {α̃_{0}, α̃_{1}, …, α̃_{n}} is a set of simple roots of Δ_{aff}.

Definition B.7 The affine Weyl group associated to (Δ, θ) is the subgroup of Aff(E, E) generated by the affine reflections sα̃ for α̃ ∈ Δ_{aff}. Alternatively,

(B.8) W_{aff} = {ϕ ∈ Aff(E, E) | ϕ factors to an element of W (Δ_{θ}) = W^{θ}}.

We now describe W_{aff}.

Definition B.9 Let

(B.10)  L_{sc} = ⟨\frac{1}{c} ∑_{k=0}^{c−1} θ^{k}(α^{∨}) | α ∈ Δ⟩

We are primarily interested in c = 1, 2, in which case:

(B.11)  L_{sc} = \{\frac{1}{2}(α^{∨} + θα^{∨}) | α ∈ Δ\}

Proposition B.12 The lattice L_{sc} is the set of translations in W_{aff}. There is an exact sequence

(B.13)(a)  0 → L_{sc} → W_{aff} → W^{θ} → 1

If we choose an element δ̃ ∈ E lying over δ we obtain a splitting of (1.18), taking W^{θ} to the the stabilizer in Aff(E) of δ̃, i.e.

(B.13)(b)  W_{aff} ≃ W^{θ} ⋉ L_{sc}

We give a few details of the map p : W_{aff} → W_{δ}. Suppose α ∈ Δ_{θ} and x ∈ ℤ. Then

p(s_{(α,x)}) = s_{α}.

Suppose c = 2, α ∈ Δ_{θ} is a short root and x ∈ ℤ + \frac{1}{2}. Then m_{α} = α^{∨}(−1) ∈ T ∩ A and

p(s_{(α,x)}) = s_{α}m_{α}

and

p(t_{\frac{1}{2} α^{∨}}) = m_{α}.

where t_{\frac{1}{2} α^{∨}} ∈ W_{aff} is translation by \frac{1}{2}α^{∨}.

References

[1] J. Adams. Lifting of characters, volume 101 of Progress in mathematics. Birkhäuser, Boston,Basel,Berlin, 1991.

[2] N. Bourbaki. Groupes et Algèbres de Lie, volume Chapitres 4,5 and 6 of Éléments de Mathématique. Masson, New York, 1981.

[3] Sigurdur Helgason. Differential geometry, Lie groups, and symmetric spaces, volume 34 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original.

[4] Anthony W. Knapp. Lie groups beyond an introduction, volume 140 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, second edition, 2002.

[5] I. G. Macdonald. Affine Hecke algebras and orthogonal polynomials, volume 157 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2003.

[6] A. L. Onishchik and È. B. Vinberg. Lie groups and algebraic groups. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites.

[7] Robert Steinberg. Endomorphisms of linear algebraic groups. Memoirs of the American Mathematical Society, No. 80. American Mathematical Society, Providence, R.I., 1968.