Journal of Lie Theory Volume 12 (2002) 41–68 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

Classification of two Involutions on Compact Semisimple Lie Groups and Root Systems

Toshihiko Matsuki

Communicated by S. Gindikin

Abstract. Let 𝔤 be a compact semisimple Lie algebra. Then we first classify pairs of involutions (σ, τ) of 𝔤 with respect to the corresponding double coset decompositions H\G/L. (Note that we don’t assume στ = τσ.) In [4], we defined a maximal torus A, a (restricted) root system Σ and a “generalized” Weyl group J and then we proved

J\A ≅ H\G/L

when G is connected. In this paper, we also compute Σ and J for some representatives of all the pairs of involutions when G is simply connected. By these data, we can compute Σ and J for “all” the pairs of involutions.

1. Introduction

Let G be a reductive Lie group with two involutions σ and τ (σ^{2} = τ^{2} = id.). Let H and L be subgroups of G such that

(G^{σ})_{0} ⊂ H ⊂ G^{σ} and (G^{τ})_{0} ⊂ L ⊂ G^{τ}.

Here G^{ρ} = {g ∈ G | ρ(g) = g} for an automorphism ρ of G and F_{0} denote the connected component of F containing the identity e for a Lie group F. In [4], we gave fundamental theorems on the structure of the double coset decompositions H\G/L.

In this paper, we consider the case that G is compact. We also assume that G is connected and semisimple for the sake of simplicity.

Let x be an element of G and ρ an automorphism of G. Then the double coset decompositions H\G/L and ρ(H)\G/xρ(L)x^{−1} are identified by the map g ↦ ρ(g)x^{−1} because

hgℓ ↦ ρ(hgℓ)x^{−1} = ρ(h)ρ(g)x^{−1}xρ(ℓ)x^{−1}

for h ∈ H and ℓ ∈ L ([4] Remark 2).

By this remark, we can define an equivalence of pairs of involutions on a Lie algebra 𝔤 as follows.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Definition 1.1. Let σ, σ′, τ and τ′ be involutions on 𝔤. Then we write

(σ, τ) ∼ (σ′, τ′)

if and only if there exist an automorphism ρ and an inner automorphism ρ_{0} of 𝔤 such that

σ′ = ρσρ^{−1} and that τ′ = ρ_{0}ρτρ^{−1}ρ_{0}^{−1}.

The first aim of this paper is to classify pairs of involutions on 𝔤 by this equivalence relation. Precise results are written in Section 2. Typical representatives will also be given for classical case. As a result, we can give the following interesting remark.

Remark 1.2. (i) When 𝔤 is of exceptional type, then we can take representatives (σ, τ) such that στ = τσ.

(ii) On the other hand, when 𝔤 is classical, there are three types of equivalence classes such that we can not take commuting pairs of involutions as representatives. (DI-III, AIII-II (p: odd, p ≠ m) and DI-I’ in Section 2. See also Remark 1 in [4].)

(iii) When 𝔤 is not simple, there are many examples as in Section 2 such that we can not take commuting pairs of involutions as representatives.

Let 𝔤 = 𝔤^{σ} ⊕ 𝔤^{−σ} = 𝔤^{τ} ⊕ 𝔤^{−τ} be the decompositions of the Lie algebra 𝔤 of G into the +1 and −1-eigenspaces for σ and τ, respectively. Let 𝔞 be a maximal abelian subspace of 𝔤^{−σ} ∩ 𝔤^{−τ} and put A = exp 𝔞. Define subgroups

N_{A} = {(h, ℓ) ∈ H × L | hAℓ^{−1} = A},
Z_{A} = {(h, ℓ) ∈ H × L | haℓ^{−1} = a for all a ∈ A}

of H × L and put J = N_{A}/Z_{A}. Then the following theorem is proved in [4].

Theorem (Theorem 1 in [4]). (i) G = HAL.

(ii) By the inclusion map, we have a bijection

J\A ≅ H\G/L.

For a linear form α : 𝔞 → iℝ (with values in pure imaginary numbers), we define the “root space”

𝔤_{ℂ}(𝔞, α) = {X ∈ 𝔤 | [Y, X] = α(Y)X for all Y ∈ 𝔞}.

Put

Σ = Σ(𝔤_{ℂ}, 𝔞) = {α ∈ i𝔞^{∗} − {0} | 𝔤_{ℂ}(𝔞, α) ≠ {0}}.

Then we have the root space decomposition

𝔤_{ℂ} = ⨁_{α∈Σ∪{0}} 𝔤_{ℂ}(𝔞, α).

Since 𝔞 ⊂ 𝔤^{στ}, we also have the following eigenspace decomposition

𝔤_{ℂ}(𝔞, α) = ⨁_{|λ|=1} 𝔤_{ℂ}(𝔞, α, λ)

where 𝔤_{ℂ}(𝔞, α, λ) = {X ∈ 𝔤_{ℂ}(𝔞, α) | στX = λX}. We proved the following in [4].

Proposition (Proposition 1 in [4]). Σ satisfies the axiom of the root systems.

Let (σ′, τ′) be another pair of involutions on 𝔤 such that σ′ = ρσρ^{−1} and that τ′ = ρ_{0}ρτρ^{−1}ρ_{0}^{−1} with some automorphism ρ and inner automorphism ρ_{0} of 𝔤. ((σ, τ) ∼ (σ′, τ′) by Definition 1.) Since ρ^{−1}ρ_{0}ρ is an inner automorphism of 𝔤, we can write ρ^{−1}ρ_{0}ρ = Ad(haℓ) with some h ∈ H, a ∈ A and ℓ ∈ L by Theorem (i). If we put ρ′ = ρAd(h), then we have

σ′ = ρ′σρ′^{−1} and τ′ = ρAd(haℓ)τAd(haℓ)^{−1}ρ^{−1} = ρ′τ_{a}ρ′^{−1}

where τ_{a} = Ad(a)τAd(a)^{−1}. Hence (σ′, τ′) is Aut (𝔤)-conjugate to (σ, τ_{a}). Let X be an element of 𝔤_{ℂ}(𝔞, α, λ). Then we have

στ_{a}X = στAd(a)^{−2}X = λa^{−2α}X = λe^{−2α(Y)}X (1.1)

if a = exp Y . On the other hand, the map

(h, ℓ) ↦ (h, aℓa^{−1})

gives a natural isomorphism of J onto the group for the pair (H, aLa^{−1}). So we have only to compute Σ, 𝔤_{ℂ}(𝔞, α, λ) and J for some representatives of the equivalence classes of (σ, τ).

The second aim of this paper is to compute Σ, dim 𝔤_{ℂ}(𝔞, α, λ) and J (when G is simply connected) for representatives (σ, τ) given in Section 2.

In Section 3, we show that the group J is determined by

Σ̃ = {(α, λ) ∈ Σ × U(1) | 𝔤_{ℂ}(𝔞, α, λ) ≠ {0}}

when G is simply connected (Proposition 3.1).

In Section 4, we give lists of Σ, dimensions of 𝔤_{ℂ}(𝔞, α, λ) and the groups J when G is connected and simply connected. (When στ = τσ, the dimensions of 𝔤_{ℂ}(𝔞, α, λ) are given in Table V of [5].)

Section 5 is an appendix for Section 3 and Section 4 to prove the following.

Corollary (of Proposition 5.2 and Proposition 5.3). When G is simply connected, the lattice {Y ∈ 𝔞 | exp Y = e} in 𝔞 is generated by the set

\{Y_{α} = \frac{4πiα}{(α, α)} | α ∈ Σ(𝔤_{ℂ}, 𝔞)\} .

Example 1.3. Let 𝔤 = 𝔲(n, 𝔽) where 𝔽 = ℝ, ℂ or ℍ. Define involutions σ and τ of 𝔤 by

σX = I_{p,q}XI_{p,q} and τX = I_{r,s}XI_{r,s}

for X ∈ 𝔤, respectively, where n = p + q = r + s, r ≥ p ≥ q ≥ s and

I_{p,q} = (\begin{matrix}I_{p} & 0 \\ 0 & −I_{q} \end{matrix}) .

Then we can take a maximal abelian subspace 𝔞 of 𝔤^{−σ} ∩ 𝔤^{−τ} of the form

𝔞 = \{Y (θ_{1}, …, θ_{s}) = (\begin{matrix}0 & & d(θ_{1}, …, θ_{s}) \\ & 0 \\ −d(θ_{1}, …, θ_{s}) & & 0 \end{matrix}) | θ_{1}, …, θ_{s} ∈ ℝ\}

where

d(θ_{1}, …, θ_{s}) = (\begin{matrix}θ_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & θ_{s} \end{matrix}) .

Define e_{j} ∈ i𝔞^{∗} by

e_{j} : Y (θ_{1}, …, θ_{s}) ↦ iθ_{j}.

Then it is wellknown that dimensions of 𝔤_{ℂ}(𝔞, α) are as follows.

\begin{matrix}α & ±e_{j} ± e_{k} & ±e_{j} & ±2e_{j} \\ dim 𝔤_{ℂ}(𝔞, α) & c & c(r − s) & c − 1 \end{matrix}

Here j ≠ k and

c = \begin{cases}1 & (𝔽 = ℝ) \\ 2 & (𝔽 = ℂ) \\ 4 & (𝔽 = ℍ). \end{cases}

Since (στ)^{2} = id., we have λ = ±1. Since 𝔤^{στ} ≅ 𝔲(p + s, 𝔽) ⊕ 𝔲(q − s, 𝔽) and 𝔤^{σ} ∩ 𝔤^{τ} ≅ 𝔲(p, 𝔽) ⊕ 𝔲(q − s, 𝔽) ⊕ 𝔲(s, 𝔽), we can easily compute dim 𝔤_{ℂ}(𝔞, α, λ) as follows.

\begin{matrix}α & ±e_{j} ± e_{k} & ±e_{j} & ±2e_{j} \\ λ = 1 & c & c(p − s) & c − 1 \\ λ = −1 & 0 & c(q − s) & 0 \end{matrix}

Let Y be an element of 𝔞 given by

Y = Y (\frac{π}{2} , …, \frac{π}{2} , 0, …, 0). (There are s_{1} \frac{π}{2} ’s and s_{2} 0’s. s_{1} + s_{2} = s.)

Put

a = exp Y = (\begin{matrix}0 & 0 & I_{s_{1}} & 0 \\ 0 & I_{r−s_{1}} & 0 & 0 \\ −I_{s_{1}} & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{s_{2}} \end{matrix} ) ∈ U(n, 𝔽).

Then we have

τ_{a}X = τAd(a)^{−2}X = I′_{r,s}XI′_{r,s}

where

I′_{r,s} = I_{r,s}a^{−2} = (\begin{matrix}−I_{s_{1}} & & 0 \\ & I_{r} \\ 0 & & −I_{s_{2}} \end{matrix} ) .

Since

a^{2e_{j}} = e^{2e_{j}(Y)} = \begin{cases}−1 & (j ≤ s_{1}) \\ 1 & (j > s_{1}), \end{cases}

we can get by (1.1) the list of dim 𝔤_{ℂ}(𝔞, α, λ) for the pair (σ, τ_{a}) of involutions as follows.

\begin{matrix}α & \begin{matrix}±e_{j} ± e_{k} \\ (j, k ≤ s_{1}) \end{matrix} & \begin{matrix}±e_{j} ± e_{k} \\ (j, k > s_{1}) \end{matrix} & \begin{matrix}±e_{j} ± e_{k} \\ (j ≤ s_{1} < k) \end{matrix} & \begin{matrix}±e_{j} \\ (j ≤ s_{1}) \end{matrix} & \begin{matrix}±e_{j} \\ (j > s_{1}) \end{matrix} & ±2e_{j} \\ λ = 1 & c & c & 0 & c(q − s) & c(p − s) & c − 1 \\ λ = −1 & 0 & 0 & c & c(p − s) & c(q − s) & 0 \end{matrix}

Remark 1.4. In this example, we choose a ∈ A so that στ_{a} = τ_{a}σ. For such pairs of involutions, dimensions of 𝔤_{ℂ}(𝔞, α, λ) (λ = ±1) are computed in [5] Table V. In the same way, we can reproduce all the data on dim 𝔤_{ℂ}(𝔞, α, λ) given in [5] from the results in Section 4.

The author is grateful to T.Oshima because the classification for exceptional case was obtained by his suggestion.

2. Classification of pairs of involutions

Let 𝔤 be a compact semisimple Lie algebra. Let G̃ = Aut (𝔤) be the group of automorphisms of 𝔤. Then G̃_{0} = Int (𝔤) is the subgroup of G̃ consisting of inner automorphisms of 𝔤. Classification of G̃-conjugacy classes of involutions on 𝔤 is well-known since it corresponds to the classification of symmetric pairs (𝔤, 𝔥). As in Section 1, we define an equivalence of pairs of involutions on 𝔤 as follows.

Definition 1.1. Let σ, σ′, τ and τ′ be involutions on 𝔤. Then we write

(σ, τ) ∼ (σ′, τ′)

if and only if there exist a ρ ∈ G̃ and ρ_{0} ∈ G̃_{0} such that

σ′ = ρσρ^{−1} and that τ′ = ρ_{0}ρτρ^{−1}ρ_{0}^{−1}.

Fix a pair of involutions (σ, τ) on 𝔤. We have only to study the equivalence in the set

S = {(σ′, τ′) | σ′ = ρσρ^{−1}, τ′ = ρ′τρ′^{−1} for some ρ, ρ′ ∈ G̃}.

Since every equivalence class in S contains an element of

S_{0} = {(σ, τ′) | τ′ = ρ′τρ′^{−1} for some ρ′ ∈ G̃},

we have only to study the equivalence of elements in S_{0}. If (σ, τ′) ∼ (σ, τ′′), then

ρσρ^{−1} = σ and ρ_{0}ρτ′ρ^{−1}ρ_{0}^{−1} = τ′′

for some ρ ∈ G̃ and ρ_{0} ∈ G̃_{0}. Write τ′ = ρ′τρ′^{−1} with a ρ′ ∈ G̃. Then

τ′′ = (ρ_{0}ρρ′)τ(ρ_{0}ρρ′)^{−1}.

There exists a natural identification between S_{0} and G̃/G̃^{τ} by the map

(σ, ρ′τρ′^{−1}) ↦ ρ′G̃^{τ}.

By this identification, (σ, τ′′) corresponds to ρ_{0}ρρ′G̃^{τ}. Hence the equivalence class in S_{0} containing τ′ is identified with the subset

G̃_{0}G̃^{σ}ρ′G̃^{τ}

of G̃. Let π be the projection

π : G̃ → G̃/G̃_{0}.

Then every equivalence class in S_{0} corresponds to a double coset

π(G̃^{σ})π(ρ′)π(G̃^{τ})

in π(G̃) with some ρ′ ∈ G̃. Hence we have only to consider the double coset decomposition

π(G̃^{σ})\π(G̃)/π(G̃^{τ}).

Suppose that 𝔤 is simple. Then it is wellknown that

π(G̃) ≅ \begin{cases}ℤ_{2} & if 𝔤 ≅ 𝔰𝔲(n) (n ≥ 3), 𝔬(2m) (m ≥ 5) or E_{6} \\ S_{3} & if 𝔤 ≅ 𝔬(8) \\ \{e\} & otherwise. \end{cases}  (2.1)

Moreover the group π(G̃^{σ}) is computed in [3] p.156 for every symmetric pair (𝔤, 𝔤^{σ}). As a result, we have π(G̃^{σ}) = π(G̃) unless

(𝔤, 𝔤^{σ}) ≅ (𝔬(4m), 𝔲(2m)) with m ≥ 3 (π(G̃) ≅ ℤ_{2}, π(G̃^{σ}) = {e})

or

(𝔤, 𝔤^{σ}) ≅ (𝔬(8), 𝔬(8 − q) ⊕ 𝔬(q)) with q = 1, 2, 3 (π(G̃) ≅ S_{3}, π(G̃^{σ}) ≅ ℤ_{2}).

Suppose that 𝔤 is classical. Then it is known that every involution σ of 𝔤 is G̃-conjugate to one of the following involutions.

type  𝔤  σ(X)  𝔥 = 𝔤^{σ}
AI  𝔰𝔲(n)  \overline{X}  𝔬(n)
AII  𝔰𝔲(2m)  J_{m}\overline{X}J_{m}^{−1}  𝔲(m, ℍ)
AIII  𝔰𝔲(n)  I_{p,q}XI_{p,q}  (𝔲(p) ⊕ 𝔲(q)) ∩ 𝔰𝔲(n)
BDI  𝔬(n)  I_{p,q}XI_{p,q}  𝔬(p) ⊕ 𝔬(q)
DIII  𝔬(2m)  J_{m}XJ_{m}^{−1}  𝔲(m)
CI  𝔲(n, ℍ)  (i_{1}I_{n})X(i_{1}I_{n})^{−1}  𝔲(n)
CII  𝔲(n, ℍ)  I_{p,q}XI_{p,q}  𝔲(p, ℍ) ⊕ 𝔲(q, ℍ)

Here p + q = n,

I_{p,q} = (\begin{matrix}I_{p} & 0 \\ 0 & −I_{q} \end{matrix}) , J_{m} = (\begin{matrix}J_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & J_{1} \end{matrix}) where J_{1} = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix})

and the field ℍ of quarternions is defined as

ℍ = {a_{0} + a_{1}i_{1} + a_{2}i_{2} + a_{3}i_{3} | a_{0}, …, a_{3} ∈ ℝ}

where i_{1}^{2} = i_{2}^{2} = i_{3}^{2} = −1, i_{1}i_{2} = −i_{2}i_{1} = i_{3}, i_{2}i_{3} = −i_{3}i_{2} = i_{1} and i_{3}i_{1} = −i_{1}i_{3} = i_{2}.

Proposition 2.1. Let (σ, τ) be a pair of involutions on a classical simple compact Lie algebra 𝔤. Then (σ, τ) or (τ, σ) is equivalent to one of the following pairs if σ and τ are not G̃_{0}-conjugate.

type  𝔤  (\begin{matrix}σ(X) \\ τ(X) \end{matrix})  (\begin{matrix}𝔥 \\ 𝔩 \end{matrix})
BDI − I  𝔬(n)  (\begin{matrix}I_{p,q}XI_{p,q} \\ I_{r,s}XI_{r,s} \end{matrix})  (\begin{matrix}𝔬(p) ⊕ 𝔬(q) \\ 𝔬(r) ⊕ 𝔬(s) \end{matrix})
AIII − III  𝔰𝔲(n)  (\begin{matrix}I_{p,q}XI_{p,q} \\ I_{r,s}XI_{r,s} \end{matrix})  (\begin{matrix}(𝔲(p) ⊕ 𝔲(q)) ∩ 𝔰𝔲(n) \\ (𝔲(r) ⊕ 𝔲(s)) ∩ 𝔰𝔲(n) \end{matrix})
CII − II  𝔲(n, ℍ)  (\begin{matrix}I_{p,q}XI_{p,q} \\ I_{r,s}XI_{r,s} \end{matrix})  (\begin{matrix}𝔲(p, ℍ) ⊕ 𝔲(q, ℍ) \\ 𝔲(r, ℍ) ⊕ 𝔲(s, ℍ) \end{matrix})
AI − III  𝔰𝔲(n)  ( \overline{X} ; I_{p,q}XI_{p,q})  (\begin{matrix}𝔬(n) \\ (𝔲(p) ⊕ 𝔲(q)) ∩ 𝔰𝔲(n) \end{matrix})
CI − II  𝔲(n, ℍ) (\begin{matrix}(i_{1}I_{n})X(i_{1}I_{n})^{−1} \\ I_{p,q}XI_{p,q} \end{matrix})  (\begin{matrix}𝔲(n) \\ 𝔲(p, ℍ) ⊕ 𝔲(q, ℍ) \end{matrix})
DI − III  𝔬(2m)  (\begin{matrix}I_{p,q}XI_{p,q} \\ J_{m}XJ_{m}^{−1} \end{matrix})  (\begin{matrix}𝔬(p) ⊕ 𝔬(q) \\ 𝔲(m) \end{matrix})
AIII − II  𝔰𝔲(2m)  (\begin{matrix}I_{p,q}XI_{p,q} \\ J_{m}\overline{X}J_{m}^{−1} \end{matrix})  (\begin{matrix}(𝔲(p) ⊕ 𝔲(q)) ∩ 𝔰𝔲(n) \\ 𝔲(m, ℍ) \end{matrix})
AI − II  𝔰𝔲(2m)  ( \overline{X} ; J_{m}\overline{X}J_{m}^{−1})  (\begin{matrix}𝔬(2m) \\ 𝔲(m, ℍ) \end{matrix})
DIII − III′  𝔬(4m)  (\begin{matrix}J_{2m}XJ_{2m}^{−1} \\ J′_{2m}XJ′_{2m}^{−1} \end{matrix})  (\begin{matrix}𝔲(2m) \\ I_{4m−1,1}𝔲(2m)I_{4m−1,1} \end{matrix})
\begin{matrix}DI − I′ \\ (q, s = 1 or 3) \end{matrix}  𝔬(8)  (\begin{matrix}I_{p,q}XI_{p,q} \\ κ^{−1}(I_{r,s}κ(X)I_{r,s}) \end{matrix})  (\begin{matrix}𝔬(p) ⊕ 𝔬(q) \\ κ(𝔬(r) ⊕ 𝔬(s)) \end{matrix})

Here n = p + q = r + s, r ≥ p ≥ q ≥ s,

J′_{2m} = (\begin{matrix}J_{2m−1} & 0 \\ 0 & −J_{1} \end{matrix})

and κ is an outer automorphism of 𝔬(8) of order 3. ( In BDI-I, AIII-III and CII-II type, we have q > s since we assume that σ and τ are not G_{0}-conjugate.)

Proof. The assertion is clear from the preceding argument except when 𝔤 = 𝔬(8). Suppose that 𝔤 = 𝔬(8). Then σ and τ are G̃-conjugate to Ad(I_{p,q}) and Ad(I_{r,s}), respectively, with some q, s = 1, 2, 3, 4. If q = 4 or s = 4, then π(G̃^{σ}) or π(G̃^{τ}) equals π(G̃) and therefore we have (σ, τ) ∼ (Ad(I_{p,q}), Ad(I_{r,s})) (BDI-I type). So we may assume that q, s = 1, 2, 3. Note that we have a set of representatives {e, π(κ)} (κ is an outer automorphism of 𝔤 of order 3) of the double coset decomposition

π(G̃^{σ})\π(G̃)/π(G̃^{τ})

since π(G̃) ≅ S_{3} and π(G̃^{σ}) ≅ π(G̃^{τ}) ≅ ℤ_{2}. If q = 2 or s = 2, then (σ, τ) or (τ, σ) is equivalent to BDI-I type or DI-III type because Ad(I_{6,2}) and Ad(J_{4}) are conjugate by an outer automorphism of 𝔤 of order 3. If q, s = 1 or 3, then (σ, τ) is equivalent to BDI-I type or DI-I’ type.

Next suppose that 𝔤 is exceptional. Since π(G̃^{σ}) = π(G̃) for every (𝔤, σ), we have only to consider all the combinations of two G̃-conjugacy classes of involutions. In Table V of [5] (c.f. Tableau II of [1]), commuting pairs of involutions are classified. But we can see that there appear “all” the combinations of involutions in that table. Thus we can take representatives (σ, τ) of pairs of involutions so that στ = τσ for exceptional case. (See the table in Section 4, B.)

Finally, suppose that 𝔤 is not simple and that 𝔤 is irreducible under the action of the pair (σ, τ), which means that there is no non-trivial ideal 𝔤_{0} of 𝔤 such that σ𝔤_{0} = τ𝔤_{0} = 𝔤_{0}. Write

𝔤 = 𝔤_{1} ⊕ ⋯ ⊕ 𝔤_{k}

with simple Lie algebras 𝔤_{j}, j = 1, …, k. By the (σ, τ)-irreducibility of 𝔤, σ𝔤_{1} or τ𝔤_{1} equals 𝔤_{j} with some j ≠ 1. Exchanging the roles of σ and τ, if necessary, and transposing 𝔤_{j} with 𝔤_{2} if j ≠ 2, we may assume that

τ𝔤_{1} = 𝔤_{2}.

If k = 2, then σ𝔤_{2} = 𝔤_{1} or 𝔤_{2}. If k ≥ 3, then we have

σ𝔤_{2} = 𝔤_{j}

with some j ≥ 3 by the (σ, τ)-irreducibility of 𝔤. Transposing 𝔤_{j} with 𝔤_{3} if j ≠ 3, we may assume that

σ𝔤_{2} = 𝔤_{3}.

Repeating this argument, we may assume that

σ𝔤_{2j} = 𝔤_{2j+1} if 2j + 1 ≤ k

and that

τ𝔤_{2j−1} = 𝔤_{2j} if 2j ≤ k.

Suppose that k is even. Then σ𝔤_{k} must equal 𝔤_{1} or 𝔤_{k}. If σ𝔤_{k} = 𝔤_{k}, then we have clearly σ𝔤_{1} = 𝔤_{1}. On the other hand, if k is odd, then we have τ𝔤_{k} = 𝔤_{k} and σ𝔤_{1} = 𝔤_{1}. Thus we have obtained the following classification of (𝔤, σ, τ) when 𝔤 is not simple.

Proposition 2.2. Suppose that 𝔤 is compact semisimple and that 𝔤 is irreducible under the action of the pair (σ, τ). Then 𝔤 is the direct sum of k-copies of a simple Lie algebra 𝔤_{1} with some k ≥ 1 and the pair (σ, τ) (or (τ, σ)) is Aut (𝔤)-conjugate to one of the following three types.

(I) k is even and

σ(X_{1}, X_{2}, …, X_{k−1}, X_{k}) = (ρ_{1}(X_{k}), X_{3}, X_{2}, …, X_{k−1}, X_{k−2}, ρ_{1}^{−1}(X_{1}))
τ(X_{1}, X_{2}, …, X_{k−1}, X_{k}) = (X_{2}, X_{1}, …, X_{k}, X_{k−1})

with some automorphism ρ_{1} of 𝔤_{1}.

(II) k is even and

σ(X_{1}, X_{2}, …, X_{k−1}, X_{k}) = (σ_{1}(X_{1}), X_{3}, X_{2}, …, X_{k−1}, X_{k−2}, τ_{1}(X_{k}))
τ(X_{1}, X_{2}, …, X_{k−1}, X_{k}) = (X_{2}, X_{1}, …, X_{k}, X_{k−1})

with some involutions σ_{1} and τ_{1} of 𝔤_{1}.

(III) k is odd and

σ(X_{1}, X_{2}, …, X_{k−1}, X_{k}) = (σ_{1}(X_{1}), X_{3}, X_{2}, …, X_{k}, X_{k−1})
τ(X_{1}, X_{2}, …, X_{k−1}, X_{k}) = (X_{2}, X_{1}, …, X_{k−1}, X_{k−2}, τ_{1}(X_{k}))

with some involutions σ_{1} and τ_{1} of 𝔤_{1}.

Remark 2.3. Suppose that (𝔤, σ, τ) is type (I).

(i) Let ρ_{0} be an inner automorphism of 𝔤_{1} and put

ρ̃_{0} = (ρ_{0}, id., …, id.) ∈ Int (𝔤).

Then we have

ρ̃_{0}σρ̃_{0}^{−1}(X_{1}, …, X_{k}) = ρ̃_{0}σ(ρ_{0}^{−1}(X_{1}), X_{2}, …, X_{k})
= ρ̃_{0}(ρ_{1}(X_{k}), X_{3}, X_{2}, …, X_{k−1}, X_{k−2}, ρ_{1}^{−1}ρ_{0}^{−1}(X_{1}))
= (ρ_{0}ρ_{1}(X_{k}), X_{3}, X_{2}, …, X_{k−1}, X_{k−2}, (ρ_{0}ρ_{1})^{−1}(X_{1})).

So we may replace ρ_{1} ∈ Aut (𝔤_{1}) by any element ρ_{0}ρ_{1} of Int (𝔤_{1})ρ_{1} since (σ, τ) ∼ (ρ̃_{0}σρ̃_{0}^{−1}, τ) by the definition of equivalence.

(ii) The space 𝔤^{−σ} ∩ 𝔤^{−τ} is written as

𝔤^{−σ} ∩ 𝔤^{−τ} = {X^{(k)} | X ∈ 𝔤_{1}^{ρ_{1}}}

where X^{(k)} = (X, −X, …, X, −X). Hence a maximal abelian subspace 𝔞 of 𝔤^{−σ} ∩ 𝔤^{−τ} is of the form

{X^{(k)} | X ∈ 𝔲_{1}}

where 𝔲_{1} is a maximal abelian subalgebra of 𝔤_{1}^{ρ_{1}} .

(iii) Let G_{1} be a connected Lie group with Lie algebra 𝔤_{1} and G the direct product of k-copies of G_{1}. Suppose that ρ_{1} lifts to an automorphism of G_{1}. Then σ and τ lift to automorphisms of G. Suppose that H = G^{σ} and L = G^{τ}. Let (x_{1}, …, x_{k}) be an element of G. Then we have

H(x_{1}, …, x_{k})L = H(x_{1}, …, x_{k})(e, …, e, x_{k}^{−1}, x_{k}^{−1})L
= H(x_{1}, …, x_{k−2}, x_{k−1}x_{k}^{−1}, e)L
= H(e, …, e, x_{k}x_{k−1}^{−1}, x_{k}x_{k−1}^{−1}, e)(x_{1}, …, x_{k−2}, x_{k−1}x_{k}^{−1}, e)L
= H(x_{1}, …, x_{k−3}, x_{k}x_{k−1}^{−1}x_{k−2}, e, e)L
= ⋯
= H(x_{1}x_{2}^{−1} ⋯ x_{k−1}x_{k}^{−1}, e, …, e)L.

Hence it is clear that the double coset decomposition H\G/L can be identified with the set

{{ρ_{1}(x)yx^{−1} | x ∈ G_{1}} | y ∈ G_{1}}

of ρ_{1}-twisted conjugacy classes in G_{1} by the map

G ∋ (x_{1}, …, x_{k}) ↦ x_{1}x_{2}^{−1} ⋯ x_{k−1}x_{k}^{−1} ∈ G_{1}.

By Theorem 1 of [4], we have

G = HAL

where A = exp 𝔞. Hence we have

G_{1} = ⋃_{x∈G_{1}} ρ_{1}(x)U_{1}x^{−1}

where U_{1} = exp 𝔲_{1}. Let G̃_{1} be a Lie group such that (G̃_{1})_{0} = G_{1} and that

ρ_{1}(g) = y^{−1}gy (g ∈ G_{1})

with some y ∈ G̃_{1}. Then we have

yG_{1} = ⋃_{x∈G_{1}} yρ_{1}(x)U_{1}x^{−1} = ⋃_{x∈G_{1}} xyU_{1}x^{−1}

(c.f. [7] Chap.II Theorem). Thus we can say that Theorem 1 of [4] includes [7] Chap.II Theorem.

Remark 2.4. Suppose that (𝔤, σ, τ) is type (II) or (III). Then the space 𝔤^{−σ} ∩ 𝔤^{−τ} is written as

𝔤^{−σ} ∩ 𝔤^{−τ} = {X^{(k)} | X ∈ 𝔤_{1}^{−σ_{1}} ∩ 𝔤_{1}^{−τ_{1}}}

where

X^{(k)} = \begin{cases}(X, −X, …, X, −X) & if k is even \\ (X, −X, …, −X, X) & if k is odd. \end{cases}

Hence a maximal abelian subspace 𝔞 of 𝔤^{−σ} ∩ 𝔤^{−τ} is of the form

{X^{(k)} | X ∈ 𝔞_{1}}

where 𝔞_{1} is a maximal abelian subspace of 𝔤_{1}^{−σ_{1}} ∩ 𝔤_{1}^{−τ_{1}} .

Let G_{1} be a connected Lie group with Lie algebra 𝔤_{1} and G the direct product of k-copies of G_{1}. Suppose that σ_{1} and τ_{1} lift to automorphisms of G_{1}. Then σ and τ lift to G. By the same argument as in Remark 2.3 (iii), we have

G^{σ}\G/G^{τ} ≅ G_{1}^{σ_{1}}\G_{1}/G_{1}^{τ_{1}}

by the map

(x_{1}, …, x_{k}) ↦ \begin{cases}x_{1}x_{2}^{−1} ⋯ x_{k−1}x_{k}^{−1} & if k is even \\ x_{1}x_{2}^{−1} ⋯ x_{k−1}^{−1}x_{k} & if k is odd. \end{cases}

3. Structure of J when G is simply connected

In this section, we assume that G is simply connected. Then it is known that

G^{ρ} is connected

for any automorphism ρ of G ([2] 3.4.Theorem). As in Section 5, we also have

{Y ∈ 𝔞 | exp Y = e} = Γ_{𝔞}

where Γ_{𝔞} is the lattice in 𝔞 generated by the set

\{Y_{α} = \frac{4πiα}{(α, α)} | α ∈ Σ(𝔤_{ℂ}, 𝔞)\} .

As in [4] Remark 3, the group J is identified with the subgroup of W_{H}(A) ▷< A (semidirect product of W_{H}(A) and A) given by

{(w, a) ∈ W_{H}(A) ▷< A | w = Ad(h)|_{𝔞} and a = hℓ^{−1} for some h ∈ N_{H}(A) and ℓ ∈ L}

where W_{S}(A) = N_{S}(A)/Z_{S}(A) for a subgroup S of G.

Let J̃ be the group of affine transformations on 𝔞 defined by

J̃ = {(w, Y) | w = Ad(h)|_{𝔞} and exp Y = hℓ^{−1} for some h ∈ N_{H}(𝔞) and ℓ ∈ L}

where (w, Y)X = wX + Y for X ∈ 𝔞. Then it is clear that

J ≅ J̃/Γ_{𝔞}

by the exponential map.

Let (α, e^{2μ}) (μ ∈ iℝ) be an element of

Σ̃ = {(α, λ) ∈ Σ(𝔤_{ℂ}, 𝔞) × U(1) | 𝔤_{ℂ}(𝔞, α, λ) ≠ {0}}.

Let X_{α} be a nonzero element of 𝔤_{ℂ}(𝔞, α, e^{2μ}) such that σX_{α} = \overline{X_{α}} (Lemma 2 of [4]) and Y an element of 𝔞 such that α(Y) = μ. Put a = exp Y and τ_{a} = Ad(a)τAd(a)^{−1} = τAd(a)^{−2}. Then we have

στ_{a}X_{α} = στAd(a)^{−2}X_{α} = X_{α}.

Hence

m_{α} = exp k(X_{α} + σX_{α}) ∈ H ∩ aLa^{−1}

defines the reflection w_{α} = Ad(m_{α})|_{𝔞} for some k ∈ ℝ. Put h = m_{α} ∈ H and ℓ = a^{−1}m_{α}a ∈ L. Then

hℓ^{−1} = m_{α}a^{−1}m_{α}^{−1}a = exp(Y − w_{α}Y) = exp(α(Y)α^{∨}) = exp μα^{∨}

where α^{∨} = 2α/(α, α) ∈ i𝔞 is the co-root of α. Hence we have

(w_{α}, μα^{∨}) ∈ J̃.

Since

(w_{α}, μα^{∨})X = w_{α}X + μα^{∨} = X − α(X)α^{∨} + μα^{∨}

for X ∈ 𝔞, (w_{α}, μα^{∨}) is the reflection in 𝔞 with respect to the hyperplane α(X) = μ.

Proposition 3.1. (i) J̃ is generated by

{(w_{α}, μα^{∨}) | (α, e^{2μ}) ∈ Σ̃}.

(ii) J is generated by

{(w_{α}, exp μα^{∨}) | (α, e^{2μ}) ∈ Σ̃}.

(iii) If W_{H∩L}(𝔞) = W (Σ) (Σ = Σ(𝔤_{ℂ}, 𝔞)), then

J = W (Σ) ▷< J_{0}

where J_{0} is the subgroup of A generated by

{exp μα^{∨} | (α, e^{2μ}) ∈ Σ̃}.

Proof. (i) Let G_{reg} = {x ∈ G | 𝔤^{−σ} ∩ 𝔤^{−τ_{x}} is abelian} be the set of regular elements in G defined in [4]. Then it is clear that

hG_{reg}ℓ^{−1} = G_{reg}

for any h ∈ H and ℓ ∈ L. Hence we have

hA_{reg}ℓ^{−1} = A_{reg}

for any (h, ℓ) ∈ N_{A} if we put

A_{reg} = A ∩ G_{reg} = {a ∈ A | a^{2α} ≠ λ for all (α, λ) ∈ Σ̃}.

Consider

𝔞_{reg} = {Y ∈ 𝔞 | exp Y ∈ A_{reg}} = {Y ∈ 𝔞 | e^{2α(Y)} ≠ λ for all (α, λ) ∈ Σ̃}.

Then we have

j𝔞_{reg} = 𝔞_{reg}  (3.1)

for any j ∈ J̃.

Let j be an element of J̃ and Δ a connected component of 𝔞_{reg} . Then by (3.1), jΔ is a connected component of 𝔞_{reg} . Let J̃_{r} denote the subgroup of J̃ generated by

{(w_{α}, μα^{∨}) | (α, e^{2μ}) ∈ Σ̃}.

Since (w_{α}, μα^{∨}) is the reflection with respect to the hyperplane {Y ∈ 𝔞 | α(Y) = μ} and since

𝔞 − 𝔞_{reg} = ⋃_{(α,e^{2μ})∈Σ̃}{Y ∈ 𝔞 | α(Y) = μ},

it is clear that there exists a j_{r} ∈ J̃_{r} such that

j_{r}jΔ = Δ.

Put j_{0} = j_{r}j = (w_{0}, Z). Then we have only to show that j_{0} = id. Let N be a positive integer such that w_{0}^{N} = id. Since

j_{0}^{N} = (w_{0}^{N}, Z′) = (id., Z′)

with some Z′ ∈ 𝔞 and since Δ is bounded, we have Z′ = 0 and therefore j_{0}^{N} = id. Let X be an element of Δ and put

X_{0} = \frac{1}{N} (X + j_{0}(X) + ⋯ + j_{0}^{N−1}(X)).

Then j_{0}(X_{0}) = X_{0} and X_{0} ∈ Δ because Δ is convex. Put x_{0} = exp X_{0}.

Let (h, ℓ) be an element of N_{A} such that Ad(h)|_{𝔞} = w_{0} and that hℓ^{−1} = exp Z. Then we have

hx_{0}ℓ^{−1} = x_{0}.

Hence

h = x_{0}ℓx_{0}^{−1} ∈ G^{σ} ∩ G^{τ_{x_{0}}} ⊂ G^{στ_{x_{0}}}.

By [2] 3.4.Theorem, G^{στ_{x_{0}}} is connected. Since x_{0} is regular, we have

𝔤^{στ_{x_{0}}} ⊂ 𝔷_{𝔤}(𝔞).

Hence we have h ∈ G^{στ_{x_{0}}} ⊂ Z_{G}(𝔞) and therefore

w_{0} = Ad(h)|_{𝔞} = id.

which implies N = 1 and j_{0} = id.

The assertions (ii) and (iii) are clear from (i).

4. Lists of Σ(𝔤_{ℂ}, 𝔞), dim 𝔤_{ℂ}(𝔞, α, λ) and J when G is simply connected

A. Classical case

We can give as follows the list of Σ = Σ(𝔤_{ℂ}, 𝔞), dim 𝔤_{ℂ}(𝔞, α, λ) and J for all (𝔤, σ, τ) given in Proposition 2.1.

\begin{matrix}type \\ 𝔤 \end{matrix}  (\begin{matrix}σ(X) \\ τ(X) \end{matrix})  Σ  dim 𝔤_{ℂ}(𝔞, α, λ)  c  \underleftrightarrow{a ∈ J_{0}}

BDI − I 𝔬(n)

AIII − III 𝔰𝔲(n) CII − II 𝔲(n, ℍ)

(\begin{matrix}I_{p,q}XI_{p,q} \\ I_{r,s}XI_{r,s} \end{matrix})

\begin{matrix}B_{s} \\ BC & s \\ BC & s \end{matrix}

(\begin{matrix}c & c(p − s) & c − 1 \\ 0 & c(q − s) & 0 \end{matrix})

1 2 4 \begin{matrix}(4.2) \\ a^{2} = e \\ a^{2} = e \end{matrix}

AI − III ; 𝔰𝔲(n) ; CI − II ; 𝔲(n, ℍ) \begin{matrix}( & \overline{X} ; I_{p,q}XI_{p,q} & ) \\ ( & \begin{matrix}(i_{1}I_{n})X(i_{1}I_{n})^{−1} \\ I_{p,q}XI_{p,q} \end{matrix} & ) \end{matrix} BC_{q} (\begin{matrix}c & c(p − q) & c − 1 \\ c & c(p − q) & c \end{matrix}) {1 \atop 2} \begin{matrix}a^{4} = e (p > q) \\ (4.3) (p = q) \end{matrix} ; a^{4} = e

DI − III ; 𝔬(2m) ; AIII − II ; 𝔰𝔲(2m)  \begin{matrix}( & \begin{matrix}I_{p,q}XI_{p,q} \\ J_{m}XJ_{m}^{−1} \end{matrix} & ) \\ ( & \begin{matrix}I_{p,q}XI_{p,q} \\ J_{m}\overline{X}J_{m}^{−1} \end{matrix} & ) \end{matrix}  BC_{q′}  \begin{matrix}( & \begin{matrix}2c & c(p − q) & 2c − 1 \\ 2c & c(p − q) & c − 1 \\ 0 & 2cε_{q} & 0 \end{matrix} & ) \\ (q′ = [q/2], ε_{q} = q − 2q′) \end{matrix} {1 \atop 2} \begin{matrix}a^{4} = e (ε_{q} = 1) \\ (4.4) (ε_{q} = 0) \end{matrix} ; a^{4} = e

\begin{matrix}AI − II \\ 𝔰𝔲(2m) \end{matrix}  ( \overline{X} ; J_{m}\overline{X}J_{m}^{−1})  A_{m−1}  (\begin{matrix}2 \\ 2 \end{matrix})  a^{4} = e

\begin{matrix}DIII − III′ \\ 𝔬(4m) \end{matrix} (\begin{matrix}J_{2m}XJ_{2m}^{−1} \\ J′_{2m}XJ′_{2m}^{−1} \end{matrix}) BC_{m−1}  (\begin{matrix}4 & 4 & 1 \\ 0 & 4 & 0 \end{matrix})  a^{2} = e

\begin{matrix}DI − I′ \\ 𝔬(8) \end{matrix} \begin{matrix}( & \begin{matrix}I_{p,q}XI_{p,q} \\ κ^{−1}(I_{r,s}κ(X)I_{r,s}) \end{matrix} & ) \\ (q, s = 1 or 3) \end{matrix} (4.1)  (\begin{matrix}1 & 1 \\ 1 & 0 \end{matrix})  (4.5)

Here the notations and assumptions for σ and τ are as in Section 2. But we choose an outer automorphism κ of 𝔬(8) of order 3 so that κAd(I_{4,4}) = Ad(I_{4,4})κ and that Ad(I′_{7,1})κAd(I′_{7,1}) = κ^{−1} where I′_{7,1} = I_{4,4}I_{5,3} (c.f. [3] p.106, p.155). The root system Σ for DI-I’ type is

Σ = \begin{cases}G_{2} & if q = s = 3 \\ ϕ & otherwise. \end{cases}  (4.1)

Matrices for dim 𝔤_{ℂ}(𝔞, α, λ) imply the following for each type of Σ. Here we write d(α, λ) = dim 𝔤_{ℂ}(𝔞, α, λ) and take a standard orthogonal basis {e_{1}, …, e_{∗}} of i𝔞^{∗} if Σ is BC_{∗} or B_{∗}-type.

BC_{∗} or B_{∗}  ( \begin{matrix}d(±e_{j} ± e_{k}, 1) & d(±e_{j}, 1) & d(±2e_{j}, 1) \\ d(±e_{j} ± e_{k}, −1) & d(±e_{j}, −1) & d(±2e_{j}, −1) \end{matrix} ; (d(±e_{j} ± e_{k}, ±i) d(±e_{j}, ±i) d(±2e_{j}, ±i)))

A_{m−1}  (\begin{matrix}d(α, 1) \\ d(α, −1) \end{matrix}) for all α ∈ Σ

G_{2}  (\begin{matrix}d(α, 1) & d(β, 1) \\ d(α, ω) & d(β, ω) \end{matrix}) (\begin{matrix}α : short roots \\ β : long roots \\ ω = (−1 ± \sqrt{3}i)/2 \end{matrix})

Since G is simply connected, we can compute the group J by Proposition 3.1 if we know Σ̃ = {(α, λ) ∈ Σ × U(1) | dim 𝔤_{ℂ}(𝔞, α, λ) > 0}. We have

J = W (Σ) ▷< J_{0}

except for AI-III type (p = q). The group J_{0} (or J) is as follows when it is not of the form {a ∈ A | a^{k} = e} with k = 2 or 4.

The group J_{0} for BDI-I type is given by

J_{0} = \{exp \frac{πi}{2} ∑_{j=1}^{s} n_{j}e_{j}^{∨} | n_{j} ∈ ℤ\}

= \{exp \frac{1}{4} ∑_{j=1}^{s} n_{j}Y_{e_{j}} | n_{j} ∈ ℤ\}

= {a ∈ A | a^{2} ∈ Z}  (4.2)

where Z is the center of G = Spin(n). (Note that |A ∩ Z| = 2.)

The group J for AI-III type (p = q) is given by

J = J^{(0)} ⊔ J^{(1)}  (4.3)

where J^{(ε)} = {(w, a) | w ∈ W^{(ε)}, a ∈ J_{0}^{(ε)}}, W (Σ) = W^{(0)}⊔W^{(1)} (W^{(0)} is D_{q} -type) and

J_{0}^{(ε)} = \{exp \frac{1}{4} ∑_{j=1}^{q} n_{j}Y_{2e_{j}} | n_{j} ∈ ℤ, ∑_{j=1}^{q} n_{j} ∈ 2ℤ + ε\} .

The group J_{0} for DI-III type (ε_{q} = 0) is given by

J_{0} = \{exp \frac{1}{4} ∑_{j=1}^{q′} n_{j}Y_{2e_{j}} | n_{j} ∈ ℤ, ∑_{j=1}^{q′} n_{j} ∈ 2ℤ\}.  (4.4)

The group J_{0} for DI-I’ type (q = s = 3) is generated by

\{exp \frac{1}{6}Y_{α} | α : short roots\}.  (4.5)

(Note that J_{0}/{a ∈ A | a^{2} = e} ≅ ℤ_{3} = ℤ/3ℤ.)

For each type of (𝔤, σ, τ), we compute 𝔤^{στ}, choose a maximal abelian subspace 𝔞 of 𝔤^{−σ} ∩ 𝔤^{−τ} and then we find Σ and dim 𝔤_{ℂ}(𝔞, α, λ) as follows. (The group J is determined by dim 𝔤_{ℂ}(𝔞, α, λ) as in Section 3 since G is simply connected.) If 𝔤 = 𝔰𝔲(n), then we may consider 𝔤 = 𝔲(n) because the center has no effect on the problems for Lie algebras.

Remark 4.1. (i) If (𝔤, σ, τ) is not of type DI-III, AIII-II (q: odd) or DI-I’, then we have chosen σ and τ in Proposition 1 so that

στ = τσ.

Hence we have (στ)^{2} = id. and therefore λ = ±1 for such cases. Although dimensions of 𝔤_{ℂ}(𝔞, α, λ) are given in Table V of [5] in these cases, we will give an explicit computation for the sake of convenience.

(ii) As is remarked in Section 1, we can easily get all the data on root multiplicities given in Table V of [5] from our two lists for classical case and exceptional case.

(iii) Type (𝔨_{ε}) in [5] corresponds to the case when σ and τ are G_{0}- conjugate.

BDI-I, AIII-III and CII-II type: For these cases, dimensions of 𝔤_{ℂ}(𝔞, α, λ) are already computed in Example 1.3. We can compute J by Proposition 3.1 if we note that

Y_{e_{j}} = \frac{4πie_{j}}{(e_{j}, e_{j})} = Y (0, …, 4π, …, 0)

where 4π is in the j -th component.

AI-III and CI-II type: Put

(𝔽, 𝔽′, η, c) = \begin{cases}(ℝ, ℂ, i, 1) & (AI-III type) \\ (ℂ, ℍ, i_{2}, 2) & (CI-II type). \end{cases}

We have 𝔤^{σ} ∩ 𝔤^{τ} = 𝔲(p, 𝔽) ⊕ 𝔲(q, 𝔽) and 𝔤^{στ} ≅ 𝔲(n, 𝔽) because στ is Aut (𝔤)- conjugate to

Ad (\begin{matrix}I_{p} & 0 \\ 0 & ηI_{q} \end{matrix}) στAd (\begin{matrix}I_{p} & 0 \\ 0 & ηI_{q} \end{matrix})^{−1}

= Ad (\begin{matrix}I_{p} & 0 \\ 0 & ηI_{q} \end{matrix}) σAd (\begin{matrix}I_{p} & 0 \\ 0 & −I_{q} \end{matrix}) Ad (\begin{matrix}I_{p} & 0 \\ 0 & ηI_{q} \end{matrix})^{−1}
= Ad (\begin{matrix}I_{p} & 0 \\ 0 & ηI_{q} \end{matrix}) σAd (\begin{matrix}I_{p} & 0 \\ 0 & ηI_{q} \end{matrix})
= σAd (\begin{matrix}I_{p} & 0 \\ 0 & −ηI_{q} \end{matrix}) Ad (\begin{matrix}I_{p} & 0 \\ 0 & ηI_{q} \end{matrix})
= σ.

(Note that σ is defined by an automorphism of 𝔽′ which maps η to −η.) We can take 𝔞 of the form

𝔞 = \{Y (θ_{1}, …, θ_{q}) = (\begin{matrix}0 & & ηd(θ_{1}, …, θ_{q}) \\ & 0 \\ ηd(θ_{1}, …, θ_{q}) & & 0 \end{matrix}) | θ_{1}, …, θ_{q} ∈ ℝ\} .

Let e_{j} be an element of i𝔞^{∗} defined by

e_{j} : Y (θ_{1}, …, θ_{q}) ↦ iθ_{j}.

Then it is clear that

dim 𝔤_{ℂ}(𝔞, ±e_{j} ± e_{k}) = 2c, dim 𝔤_{ℂ}(𝔞, ±e_{j}) = 2c(p − q), dim 𝔤_{ℂ}(𝔞, ±2e_{j}) = 2c − 1

and that 𝔤_{ℂ}(𝔞, α) = {0} for other α ∈ i𝔞^{∗} − {0}. On the other hand, we have

dim 𝔤_{ℂ}(𝔞, ±e_{j} ± e_{k}, 1) = c, dim 𝔤_{ℂ}(𝔞, ±e_{j}, 1) = c(p − q) and dim 𝔤_{ℂ}(𝔞, ±2e_{j}, 1) = c − 1

since 𝔤^{στ}/𝔤^{σ} ∩ 𝔤^{τ} ≅ 𝔲(n, 𝔽)/𝔲(p, 𝔽) ⊕ 𝔲(q, 𝔽).

DI-III and AIII-II type: Put

(𝔽, 𝔽′, c) = \begin{cases}(ℝ, ℂ, 1) & (DI-III type) \\ (ℂ, ℍ, 2) & (AIII-II type). \end{cases}

First suppose that q is even. Then 𝔤^{σ} ∩ 𝔤^{τ} ≅ 𝔲(p′, 𝔽′) ⊕ 𝔲(q′, 𝔽′) (p′ = p/2, q′ = q/2) and 𝔤^{στ} ≅ 𝔤^{τ} ≅ 𝔲(m, 𝔽′) because I_{p,q}J_{m} is O(2m)-conjugate to J_{m}. We can take a maximal abelian subspace 𝔞 in 𝔤^{−σ} ∩ 𝔤^{−τ} of the form 𝔞 =

\{Y (θ_{1}, …, θ_{q′}) = (\begin{matrix}0 & & d(θ_{1}I′_{2}, …, θ_{q′}I′_{2}) \\ & 0 \\ −d(θ_{1}I′_{2}, …, θ_{q′}I′_{2}) & & 0 \end{matrix}) | θ_{1}, …, θ_{q′} ∈ ℝ\}

(4.6), where

I′_{2} = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) and d(A_{1}, …, A_{q′}) = (\begin{matrix}A_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & A_{q′} \end{matrix} )

for matrices A_{1}, …, A_{q′}. Let e_{j} be an element of i𝔞^{∗} defined by

e_{j} : Y (θ_{1}, …, θ_{q′}) ↦ iθ_{j}.

Take an abelian subspace

𝔟 = \{Z(φ_{1}, …, φ_{q}) = (\begin{matrix}0 & & d′(φ_{1}, …, φ_{q}) \\ & 0 \\ −^{t}d′(φ_{1}, …, φ_{q}) & & 0 \end{matrix}) | φ_{1}, …, φ_{q} ∈ ℝ\}

of 𝔤 containing 𝔞, where d′(φ_{1}, …, φ_{q}) = d(φ_{1}, …, φ_{q})d(I′_{2}, …, I′_{2}). Define f_{j} ∈ i𝔟^{∗} by

f_{j} : Z(φ_{1}, …, φ_{q}) ↦ iφ_{j}.

Then we have

f_{2j−1}|_{𝔞} = f_{2j}|_{𝔞} = e_{j}

for j = 1, …, q′. So we have

dim 𝔤_{ℂ}(𝔞, ±e_{j} ± e_{k}) = dim 𝔤_{ℂ}(𝔟, ±f_{2j−1} ± f_{2k−1}) + dim 𝔤_{ℂ}(𝔟, ±f_{2j−1} ± f_{2k})
+ dim 𝔤_{ℂ}(𝔟, ±f_{2j} ± f_{2k−1}) + dim 𝔤_{ℂ}(𝔟, ±f_{2j} ± f_{2k})
= 4c,
dim 𝔤_{ℂ}(𝔞, ±e_{j}) = dim 𝔤_{ℂ}(𝔟, ±f_{2j−1}) + dim 𝔤_{ℂ}(𝔟, ±f_{2j})
= 2c(p − q),
dim 𝔤_{ℂ}(𝔞, ±2e_{j}) = dim 𝔤_{ℂ}(𝔟, ±2f_{2j−1}) + dim 𝔤_{ℂ}(𝔟, ±2f_{2j})
+ dim 𝔤_{ℂ}(𝔟, ±(f_{2j−1} + f_{2j}))
= 2(c − 1) + c
= 3c − 2.

On the other hand, we have

dim 𝔤_{ℂ}(𝔞, ±e_{j} ± e_{k}, 1) = 2c, dim 𝔤_{ℂ}(𝔞, ±e_{j}, 1) = c(p − q) and dim 𝔤_{ℂ}(𝔞, ±2e_{j}, 1) = 2c − 1

since 𝔤^{στ}/𝔤^{σ} ∩ 𝔤^{τ} ≅ 𝔲(m, 𝔽′)/𝔲(p′, 𝔽′) ⊕ 𝔲(q′, 𝔽′).

Next suppose that q is odd. Then we have

𝔤^{(στ)^{2}} ≅ 𝔲(2m − 2, 𝔽) ⊕ 𝔲(2, 𝔽)

because (στ)^{2}(X) = (I_{p,q}J_{m})^{2}X(I_{p,q}J_{m})^{−2} and

(I_{p,q}J_{m})^{2} = (\begin{matrix}J_{p′} & & 0 \\ & I′_{2} \\ 0 & & −J_{q′} \end{matrix} )^{2} = (\begin{matrix}−I_{2p′} & & 0 \\ & I_{2} \\ 0 & & −I_{2q′} \end{matrix} )

where p′ = [p/2]. Moreover we have

𝔤^{(στ)^{2}} ∩ 𝔤^{σ} ≅ 𝔲(2p′, 𝔽) ⊕ 𝔲(2q′, 𝔽) ⊕ 𝔲(1, 𝔽) ⊕ 𝔲(1, 𝔽)

and

𝔤^{(στ)^{2}} ∩ 𝔤^{τ} ≅ 𝔲(m − 1, 𝔽′) ⊕ 𝔲(1, 𝔽′).

Since 𝔞 ⊂ 𝔤^{−σ} ∩𝔤^{−τ} ⊂ 𝔤^{στ} ⊂ 𝔤^{(στ)^{2}} and since 𝔲(2, 𝔽) = (𝔲(1, 𝔽)⊕𝔲(1, 𝔽))+𝔲(1, 𝔽′), we can choose 𝔞 of the same form as (4.6). So we can compute dim 𝔤_{ℂ}(𝔞, α) and dim 𝔤_{ℂ}(𝔞, α, ±1) for all α ∈ Σ by the results when q is even. Especially we have

dim 𝔤_{ℂ}^{(στ)^{2}}(𝔞, ±e_{j}) = 2c(2p′ − 2q′),
dim 𝔤_{ℂ}(𝔞, ±e_{j}) = 2c(2m − 4q′) = 2c(2p′ − 2q′ + 2)

and dim 𝔤_{ℂ}^{(στ)^{2}}(𝔞, α) = dim 𝔤_{ℂ}(𝔞, α) if α ≠ ±e_{j}. Hence we have

dim 𝔤_{ℂ}(𝔞, α, ±i) = \begin{cases}2c & if α = ±e_{j} \\ 0 & otherwise. \end{cases}

AI-II type: Since στ(X) = J_{m}XJ_{m}^{−1} and since J_{m} is U(2m)-conjugate to

(\begin{matrix}iI_{m} & 0 \\ 0 & −iI_{m} \end{matrix}) ,

we have 𝔤^{στ} ≅ 𝔲(m) ⊕ 𝔲(m). On the other hand, we have 𝔤^{σ} ∩ 𝔤^{τ} = {X ∈ 𝔬(2m) | J_{m}XJ_{m}^{−1} = X} ≅ 𝔲(m). We can take a maximal abelian subspace 𝔞 in 𝔤^{−σ} ∩ 𝔤^{−τ} of the form

𝔞 = {Y (θ_{1}, …, θ_{m}) = d(iθ_{1}I_{2}, …, iθ_{m}I_{2}) | θ_{1}, …, θ_{m} ∈ ℝ}.

Let e_{j} be an element of i𝔞^{∗} defined by

e_{j} : Y (θ_{1}, …, θ_{m}) ↦ iθ_{j}.

Take an abelian subspace

𝔟 = {Z(φ_{1}, …, φ_{2m}) = d(iφ_{1}, …, iφ_{2m}) | φ_{1}, …, φ_{2m} ∈ ℝ}

of 𝔤 containing 𝔞 and define f_{j} ∈ i𝔟^{∗} by

f_{j} : Z(φ_{1}, …, φ_{q}) ↦ iφ_{j}.

Then we have

f_{2j−1}|_{𝔞} = f_{2j}|_{𝔞} = e_{j}

for j = 1, …, m. So we have

dim 𝔤_{ℂ}(𝔞, e_{j} − e_{k}) = dim 𝔤_{ℂ}(𝔟, f_{2j−1} − f_{2k−1}) + dim 𝔤_{ℂ}(𝔟, f_{2j−1} − f_{2k})
+ dim 𝔤_{ℂ}(𝔟, f_{2j} − f_{2k−1}) + dim 𝔤_{ℂ}(𝔟, f_{2j} − f_{2k})
= 4

and 𝔤_{ℂ}(𝔞, α) = {0} for other α ∈ i𝔞^{∗} − {0}. On the other hand, we have dim 𝔤_{ℂ}(𝔞, e_{j} − e_{k}, 1) = 2 since 𝔤^{στ}/𝔤^{σ} ∩ 𝔤^{τ} ≅ 𝔲(m) ⊕ 𝔲(m)/{(X, X) | X ∈ 𝔲(m)}.

DIII-III’ type: We have 𝔤^{στ} ≅ 𝔬(4m − 2) ⊕ 𝔬(2) and 𝔤^{σ} ∩ 𝔤^{τ} ≅ 𝔲(2m − 1) ⊕ 𝔲(1). We can take a maximal abelian subspace 𝔞 in 𝔤^{−σ} ∩ 𝔤^{−τ} consisting of the elements

Y (θ_{1}, …, θ_{m−1}) = (\begin{matrix}0 & d(θ_{1}I′_{2}, …, θ_{m−1}I′_{2}) & 0 \\ −d(θ_{1}I′_{2}, …, θ_{m−1}I′_{2}) & 0 & 0 \\ 0 & 0 & 0 \end{matrix})

where θ_{1}, …, θ_{m−1} ∈ ℝ. Let e_{j} be an element of i𝔞^{∗} defined by

e_{j} : Y (θ_{1}, …, θ_{m−1}) ↦ iθ_{j}.

Since the symmetric pair (𝔤^{στ}, 𝔤^{σ} ∩ 𝔤^{τ}) is DIII-type, we have dim 𝔤_{ℂ}(𝔞, ±e_{j}, 1) = dim 𝔤_{ℂ}(𝔞, ±e_{j} ± e_{k}, 1) = 4 and dim 𝔤_{ℂ}(𝔞, ±2e_{j}, 1) = 1. We also have easily that

dim 𝔤_{ℂ}(𝔞, α, −1) = \begin{cases}4 & if α = ±e_{j} \\ 0 & otherwise. \end{cases}

(We may take an abelian subspace 𝔟 of 𝔤 as in DI-III and AIII-II type.)

DI-I’ type: Remark that G = HL if and only if G = H′L′ with some Int (𝔤)-conjugates H′ and L′ of H and L, respectively.

First we consider the case q = s = 1 and show that G = HL. Put σ′ = Ad(I′_{7,1}) and τ′ = κAd(I′_{7,1})κ^{−1}. Then we have only to show that G = G^{σ′}G^{τ′} by the above remark. We have

σ′τ′ = σ′κ^{−1}σ′κ = κ^{2} = κ^{−1}

and therefore 𝔤^{σ′τ′} = 𝔤^{κ} is G_{2}-type. Since κ and σ′ generate a subgroup of Aut (𝔤) isomorphic to S_{3}, it is known that

𝔤^{κ} ⊂ 𝔤^{σ′}.

Hence we have 𝔤^{σ′τ′} ⊂ 𝔤^{σ′} and therefore

𝔤^{−σ′} ∩ 𝔤^{−τ′} = 𝔤^{σ′τ′} ∩ 𝔤^{−σ′} = {0},

which implies G = G^{σ′}G^{τ′}.

Next we consider the case (q, s) = (3, 1) and show that G = HL. Put τ′ = κ^{−1}Ad(I′_{7,1})κ. Then we have only to show that

𝔤^{−σ} ∩ 𝔤^{−τ′} = {0}

as in the case of q = s = 1. Since I_{5,3} = I_{4,4}I′_{7,1}, we have

στ′ = Ad(I_{4,4})Ad(I′_{7,1})κ^{−1}Ad(I′_{7,1})κ = Ad(I_{4,4})κ^{−1}

and (στ′)^{2} = κ^{−2} = κ because κAd(I_{4,4}) = Ad(I_{4,4})κ. Hence 𝔤^{στ′} ⊂ 𝔤^{(στ′)^{2}} = 𝔤^{κ} ⊂ 𝔤^{τ′} and therefore

𝔤^{−σ} ∩ 𝔤^{−τ′} = 𝔤^{στ′} ∩ 𝔤^{−τ′} = {0}.

Finally, consider the case q = s = 3. We have

στ = Ad(I_{5,3})κ^{−1}Ad(I_{5,3})κ
= Ad(I_{4,4})Ad(I′_{7,1})κ^{−1}Ad(I_{4,4})Ad(I′_{7,1})κ
= Ad(I′_{7,1})κ^{−1}Ad(I′_{7,1})κ
= κ^{−1}

since κAd(I_{4,4}) = Ad(I_{4,4})κ. Hence 𝔤^{στ} = 𝔤^{κ} is G_{2}-type and 𝔤^{σ} ∩ 𝔤^{τ} = 𝔤^{σ} ∩ 𝔤^{στ} ≅ 𝔰𝔲(2) ⊕ 𝔰𝔲(2) since (𝔤^{στ}, 𝔤^{σ} ∩ 𝔤^{τ}) is a nontrivial symmetric pair. So we can choose a two-dimensional maximal abelian subspace 𝔞 of 𝔤^{−σ} ∩ 𝔤^{−τ}. It is clear that dimensions of 𝔤_{ℂ}(𝔞, α, λ) are as in the list.

B. Exceptional case

When 𝔤 is exceptional, we can give as follows the list of Σ, dim 𝔤_{ℂ}(𝔞, α, λ) and J for representatives of all the triples (𝔤, σ, τ) such that σ and τ are not G_{0}- conjugate. We may assume that στ = τσ and that (𝔤^{στ}, 𝔥 ∩ 𝔩) are as follows (c.f. Table V in [5]). Dimensions of 𝔤_{ℂ}(𝔞, α, λ) are given in Table V in [5]. (We can easily compute them by the same arguments as in the classical cases.) By Proposition 3.1, the group J is determined by Σ̃.

\begin{matrix}type \\ 𝔤 \end{matrix}  \begin{matrix}𝔥 \\ 𝔩 \end{matrix}  \begin{matrix}𝔤^{στ} \\ 𝔥 ∩ 𝔩 \end{matrix}  Σ dim 𝔤_{ℂ}(𝔞, α, λ) \underleftrightarrow{a ∈ J_{0}}

EI − II E_{6}

EV − VI E_{7}

EVIII − IX E_{8}

\begin{matrix}𝔰𝔭(4) \\ 𝔰𝔲(6) ⊕ 𝔰𝔲(2) \\ 𝔰𝔲(8) \\ 𝔬(12) ⊕ 𝔰𝔲(2) \\ 𝔬(16) \\ E_{7} ⊕ 𝔰𝔲(2) \end{matrix} \begin{matrix}F_{4} \\ 𝔰𝔭(3) ⊕ 𝔰𝔭(1) \\ E_{6} ⊕ ℝ \\ 𝔰𝔲(6) ⊕ 𝔰𝔲(2) ⊕ ℝ \\ E_{7} ⊕ 𝔰𝔲(2) \\ 𝔬(12) ⊕ 𝔬(4) \end{matrix}

F_{4}

\begin{matrix}( & 1 1 1 0 & ) \\ ( & 2 1 2 0 & ) \\ ( & 4 1 4 0 & ) \end{matrix}

(4.7)

\begin{matrix}EV − VII \\ E_{7} \end{matrix}  \begin{matrix}𝔰𝔲(8) \\ E_{6} ⊕ ℝ \end{matrix}  \begin{matrix}𝔬(12) ⊕ 𝔰𝔲(2) \\ 𝔰𝔲(6) ⊕ 𝔰𝔲(2) ⊕ ℝ \end{matrix} C_{3}  (\begin{matrix}4 & 0 & 1 \\ 4 & 0 & 0 \end{matrix})  (4.4)

EII − III ; E_{6} ; EVI − VII ; E_{7} \begin{matrix}𝔰𝔲(6) ⊕ 𝔰𝔲(2) \\ 𝔬(10) ⊕ ℝ \\ 𝔬(12) ⊕ 𝔰𝔲(2) \\ E_{6} ⊕ ℝ \end{matrix}  \begin{matrix}𝔬(10) ⊕ ℝ \\ 𝔰𝔲(5) ⊕ ℝ^{2} \\ E_{6} ⊕ ℝ \\ 𝔬(10) ⊕ ℝ^{2} \end{matrix}  BC_{2}  \begin{matrix}( & 4 4 1 2 4 0 & ) \\ ( & 6 8 1 2 8 0 & ) \end{matrix}  (4.4)

\begin{matrix}EI − III \\ E_{6} \end{matrix}  \begin{matrix}𝔰𝔭(4) \\ 𝔬(10) ⊕ ℝ \end{matrix}  \begin{matrix}𝔰𝔭(4) \\ 𝔰𝔭(2) ⊕ 𝔰𝔭(2) \end{matrix} BC_{2} (\begin{matrix}3 & 4 & 0 \\ 3 & 4 & 1 \end{matrix}) a^{4} = e

\begin{matrix}EI − IV \\ E_{6} \end{matrix}  \begin{matrix}𝔰𝔭(4) \\ F_{4} \end{matrix}  \begin{matrix}𝔰𝔲(6) ⊕ 𝔰𝔲(2) \\ 𝔰𝔭(3) ⊕ 𝔰𝔭(1) \end{matrix} A_{2}  (\begin{matrix}4 \\ 4 \end{matrix})  a^{4} = e

\begin{matrix}EII − IV \\ E_{6} \end{matrix}  \begin{matrix}𝔰𝔲(6) ⊕ 𝔰𝔲(2) \\ F_{4} \end{matrix} \begin{matrix}𝔰𝔭(4) \\ 𝔰𝔭(3) ⊕ 𝔰𝔭(1) \end{matrix} BC_{1}  (\begin{matrix}8 & 3 \\ 8 & 5 \end{matrix}) a^{4} = e

\begin{matrix}EIII − IV \\ E_{6} \end{matrix}  \begin{matrix}𝔬(10) ⊕ ℝ \\ F_{4} \end{matrix}  \begin{matrix}F_{4} \\ 𝔬(9) \end{matrix}  BC_{1}  (\begin{matrix}8 & 7 \\ 8 & 1 \end{matrix}) a^{4} = e

\begin{matrix}FI − II \\ F_{4} \end{matrix}  \begin{matrix}𝔰𝔭(3) ⊕ 𝔰𝔭(1) \\ 𝔬(9) \end{matrix} \begin{matrix}𝔰𝔭(3) ⊕ 𝔰𝔭(1) \\ 𝔬(5) ⊕ 𝔬(4) \end{matrix} BC_{1}  (\begin{matrix}4 & 3 \\ 4 & 4 \end{matrix}) a^{4} = e

Here the matrices for dim 𝔤_{ℂ}(𝔞, α, λ) are the same as in the classical case if Σ is A_{2}, BC_{2} or C_{3}-type. If Σ is F_{4}-type or BC_{1}-type, then the matrix implies

(\begin{matrix}dim 𝔤_{ℂ}(𝔞, α, 1) & dim 𝔤_{ℂ}(𝔞, β, 1) \\ dim 𝔤_{ℂ}(𝔞, α, −1) & dim 𝔤_{ℂ}(𝔞, β, −1) \end{matrix})

for short roots α and long roots β. If Σ is F_{4}-type, then J_{0} is generated by

\{exp \frac{1}{4}Y_{α} | α : short roots\}.  (4.7)

(Note that J_{0}/{a ∈ A | a^{2} = e} ≅ ℤ_{2} × ℤ_{2}.)

C. The case 𝔤 is not simple

Type I: Let (𝔤, σ, τ) be as in Proposition 2.2 (I) and take a maximal abelian subspace 𝔞 of 𝔤^{−σ} ∩ 𝔤^{−τ} as in Remark 2.3 (ii). Then the space 𝔤_{ℂ}(𝔞, α, λ) consists of the elements

(X_{1}, X_{2}, λ^{−1}X_{1}, λX_{2}, …, λ^{−m+1}X_{1}, λ^{m−1}X_{2})

with X_{1} ∈ (𝔤_{1})_{ℂ}(𝔲_{1}, α_{1}, λ^{m}) and X_{2} ∈ (𝔤_{1})_{ℂ}(𝔲_{1}, −α_{1}, λ^{−m}). Here m = k/2, α_{1}(Y) = α(Y^{(k)}) for Y ∈ 𝔲_{1} and (𝔤_{1})_{ℂ}(𝔲_{1}, β, μ) = {X ∈ (𝔤_{1})_{ℂ}(𝔲_{1}, β) | ρ_{1}X = μX} for (β, μ) ∈ i𝔲_{1}^{∗} × U(1).

Let G_{1}, G, H, L, A and U_{1} be as in Remark 2.3 (iii). Let (h, ℓ) be an element of N_{A}. Write h = (h_{1}, …, h_{k}) and ℓ = (ℓ_{1}, …, ℓ_{k}) where h_{1}, …, h_{k}, ℓ_{1}, …, ℓ_{k} ∈ G_{1}. Since hℓ^{−1} ∈ A, we can write

(h_{1}ℓ_{1}^{−1}, …, h_{k}ℓ_{k}^{−1}) = hℓ^{−1} = a^{(k)} = (a, a^{−1}, …, a, a^{−1})

with some a ∈ U_{1}. Hence we have

ℓ_{k−1} = ℓ_{k} = ah_{k}
h_{k−2} = h_{k−1} = a^{2}h_{k}
ℓ_{k−3} = ℓ_{k−2} = a^{3}h_{k}
⋯
ρ_{1}(h_{k}) = h_{1} = a^{k}h_{k}

by the definition of H and L. Thus we can identify the group J with the group

{(Ad(g)|_{U_{1}}, b) ∈ W_{G_{1}}(U_{1}) ▷< U_{1} | ρ_{1}(g)g^{−1} = b^{k}}

by the map

(h, ℓ) ↦ (Ad(h_{k})|_{U_{1}}, a).

(Note that A is identified with U_{1} by a^{(k)} ↦ a.)

Define subgroups

N_{1} = {g ∈ G_{1} | ρ_{1}(g)U_{1}g^{−1} = U_{1}}, Z_{1} = {g ∈ G_{1} | ρ(g)ag^{−1} = a for all a ∈ U_{1}}

of G_{1} and put J_{1} = N_{1}/Z_{1}. Then by the same argument as in [4] Remark 3, we can identify J_{1} with the group

{(Ad(g)|_{U_{1}}, b) ∈ W_{G_{1}}(U_{1}) ▷< U_{1} | ρ_{1}(g)g^{−1} = b}.

Comparing J and J_{1}, we can see that the map A ∋ a^{(k)} ↦ a^{k} ∈ U_{1} induces a bijection

J\A ≅ J_{1}\U_{1}.

Summarizing the above arguments, we have the following natural commuting diagram of bijections.

J\A ⟶  J_{1}\U_{1}
↓  ↓
H\G/L ⟶ G_{1}/ρ_{1}-twisted conj.

Here the vertical arrows are given by the inclusions and the horizontal ones are given by the map

(x_{1}, …, x_{k}) ↦ x_{1}x_{2}^{−1} ⋯ x_{k−1}x_{k}^{−1}.

Note that the restriction of this map to A is a^{(k)} ↦ a^{k}.

If ρ_{1} is an inner automorphism, then we may assume that ρ_{1} = id. by Remark 2.3 (i). If G_{1} is simply connected, then it is known that J_{1} ≅ W_{G_{1}}(U_{1}). Hence

J ≅ W_{G_{1}}(U_{1}) ▷< {a ∈ U_{1} | a^{k} = e}.

Of course, this follows also from Proposition 3.1.

On the other hand, if ρ_{1} is an outer automorphism, then we may assume that (𝔤_{1}, 𝔤_{1}^{ρ_{1}}) is one of the following five types by Remark 2.3 (i) and (2.1).

𝔤_{1}  𝔤_{1}^{ρ_{1}}  Σ((𝔤_{1})_{ℂ}, 𝔲_{1}) dim(𝔤_{1})_{ℂ}(𝔲_{1}, α, λ) J_{0} (or J) when k = 2
𝔰𝔲(2m)  𝔰𝔭(m)  C_{m}  (\begin{matrix}1 & 0 & 1 \\ 1 & 0 & 0 \end{matrix})  (4.4)
𝔰𝔲(2m + 1) 𝔬(2m + 1)  BC_{m}  (\begin{matrix}1 & 1 & 0 \\ 1 & 1 & 1 \end{matrix})  (4.3)
𝔬(2m) 𝔬(2m − 1)  B_{m−1}  (\begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \end{matrix})  (4.2)
E_{6}  F_{4}  F_{4}  (\begin{matrix}1 & 1 \\ 1 & 0 \end{matrix})  (4.7)
𝔬(8)  G_{2}  G_{2}  (\begin{matrix}1 & 1 \\ 1 & 0 \end{matrix})  (4.5)

Here the matrices for dim(𝔤_{1})_{ℂ}(𝔲_{1}, α, λ) are as in the simple cases. If k = 2, then we can see that every Σ̃ = {(α, λ) ∈ i𝔞^{∗} × U(1) | 𝔤_{ℂ}(𝔞, α, λ) ≠ {0}} is equal to one of the classical cases. So the group J is the same by Proposition 3.1.

Type (II) and (III): Let (𝔤, σ, τ) be as in Proposition 2.2 (II) and (III). Take a maximal abelian subspace 𝔞 of 𝔤^{−σ} ∩ 𝔤^{−τ} as in Remark 2.4. If k = 2m is even, then the space 𝔤_{ℂ}(𝔞, α, λ) consists of the elements

(X, λ^{−2m+1}τ_{1}X, λ^{−1}X, λ^{−2m+2}τ_{1}X, …, λ^{−m+2}X, λ^{−m−1}τ_{1}X, λ^{−m+1}X, λ^{−m}τ_{1}X)

with X ∈ (𝔤_{1})_{ℂ}(𝔞_{1}, α_{1}, λ^{k}). Here α_{1}(Y) = α(Y^{(k)}) for Y ∈ 𝔞_{1} and (𝔤_{1})_{ℂ}(𝔞_{1}, β, μ) = {(X ∈ (𝔤_{1})_{ℂ}(𝔞_{1}, β) | σ_{1}τ_{1}X = μX} for (β, μ) ∈ i𝔞_{1}^{∗}×U(1). Similarly, if k = 2m+1 is odd, then the space 𝔤_{ℂ}(𝔞, α, λ) consists of the elements

(X, λ^{−2m}τ_{1}X, λ^{−1}X, λ^{−2m+1}τ_{1}X, …, λ^{−m−2}τ_{1}X, λ^{−m+1}X, λ^{−m−1}τ_{1}X, λ^{−m}X)

with X ∈ (𝔤_{1})_{ℂ}(𝔞_{1}, α_{1}, λ^{k}).

Let G_{1}, G, A and A_{1} be as in Remark 2.4. Let J_{1} be the group for (G_{1}, G_{1}^{σ_{1}}, G_{1}^{τ_{1}}). Then by a similar argument as in type I case, we have the following natural commuting diagram of bijections.

J\A  ⟶  J_{1}\A_{1}
↓  ↓
G^{σ}\G/G^{τ} ⟶ G_{1}^{σ_{1}}\G_{1}/G_{1}^{τ_{1}}

Here the vertical arrows are given by the inclusions and the horizontal ones are given by the map

(x_{1}, …, x_{k}) ↦ \begin{cases}x_{1}x_{2}^{−1} ⋯ x_{k−1}x_{k}^{−1} & if k is even \\ x_{1}x_{2}^{−1} ⋯ x_{k−1}^{−1}x_{k} & if k is odd. \end{cases}

Note that the restriction of this map to A is a^{(k)} ↦ a^{k}.

5. Appendix

Let ρ be an automorphism of 𝔤. Take a maximal abelian subalgebra 𝔲 of 𝔤^{ρ} and let 𝔱 denote the centralizer of 𝔲 in 𝔤. Then the following lemma seems to be known. For the sake of completeness, we will give a proof by the same argument as in the proof of [2] 3.4.Theorem.

Lemma 5.1. 𝔱 is abelian.

Proof. Let 𝔷 be the center of 𝔱 and 𝔰 = [𝔱, 𝔱] the semisimple part of 𝔱. Then

𝔱 = 𝔷 ⊕ 𝔰.

Suppose that 𝔰 ≠ {0}. Then we will get a contradiction.

Since 𝔲 is ρ-stable, 𝔰 is also ρ-stable. By [7] Chap.II Section 2, we have 𝔰^{ρ} ≠ {0} and therefore 𝔰^{ρ} contains a nontrivial abelian subalgebra 𝔲_{𝔰}. Thus we get an abelian subalgebra 𝔲 ⊕ 𝔲_{𝔰} of 𝔤^{ρ}. But this contradicts to the assumption that 𝔲 is maximal abelian in 𝔤^{ρ}.

Let Γ_{∗} (∗ = 𝔱 or 𝔲) denote the lattices in ∗ generated by the set

\{Y_{α} = \frac{4πiα}{(α, α)} | α ∈ Σ(𝔤_{ℂ}, ∗)\} .  (5.1)

Let G be the connected simply connected Lie group with Lie algebra 𝔤. Then it is known that

{Y ∈ 𝔱 | exp Y = e} = Γ_{𝔱}.

So the following proposition implies that

{Y ∈ 𝔲 | exp Y = e} = Γ_{𝔲}.

Proposition 5.2. Γ_{𝔲} = Γ_{𝔱} ∩ 𝔲.

Proof. Since 𝔤 is semisimple, 𝔤 is written as a direct sum

𝔤 = 𝔤_{1} ⊕ ⋯ ⊕ 𝔤_{k}

of simple Lie algebras 𝔤_{1}, …, 𝔤_{k}. Considering each irreducible component of the ρ-action on 𝔤, we may assume that 𝔤_{1} = 𝔤_{2} = ⋯ = 𝔤_{k} and that

ρ(X_{1}, …, X_{k}) = (ρ_{1}(X_{k}), X_{1}, …, X_{k−1})

with some automorphism ρ_{1} of 𝔤_{1}. Let X = (X_{1}, …, X_{k}) be an element of 𝔤^{ρ}. Then we have

(ρ_{1}(X_{k}), X_{1}, …, X_{k−1}) = (X_{1}, …, X_{k})

and therefore ρ_{1}(X_{k}) = X_{1} = X_{2} = ⋯ = X_{k} ∈ 𝔤_{1}^{ρ_{1}} . Hence we have shown that

𝔤^{ρ} = {(X, …, X) ∈ 𝔤 | X ∈ 𝔤_{1}^{ρ_{1}}} ≅ 𝔤_{1}^{ρ_{1}}.

Thus the space 𝔲 is of the form

𝔲 = {(X, …, X) ∈ 𝔤 | X ∈ 𝔲_{1}}

where 𝔲_{1} is a maximal abelian subalgebra of 𝔤_{1}^{ρ_{1}} . So we have only to consider the case that 𝔤 is simple in the following.

Let W̃(∗) = W (∗) ▷< Γ_{∗} be the group of affine transformations on the spaces ∗ = 𝔱 or 𝔲 generated by reflections

w_{α,n} : Y ↦ w_{α}(Y) + nY_{α} = Y − α(Y)\frac{2α}{(α, α)} + nY_{α}

with respect to the hyperplanes {Y ∈ ∗ | α(Y) = 2πin} for α ∈ Σ(∗) and n ∈ ℤ. We can define the set C(∗) of “Weyl chambers” in ∗ = 𝔱, 𝔲 consisting of connected components of the set

{Y ∈ ∗ | α(Y) ≠ 2πin for all α ∈ Σ(∗) and n ∈ ℤ}.

Then it is known that W̃(∗) acts simply transitively on C(∗). Note that every Δ_{𝔲} ∈ C(𝔲) can be written as

Δ_{𝔲} = Δ_{𝔱} ∩ 𝔲

with a unique Δ_{𝔱} ∈ C(𝔱) because there is no root α ∈ Σ(𝔱) such that α|_{𝔲} = 0 by Lemma 5.1.

Put W̃(𝔱)_{𝔲} = {w ∈ W̃(𝔱) | w(𝔲) = 𝔲}|_{𝔲}. Then we have only to show that

W̃(𝔱)_{𝔲} = W̃(𝔲)  (5.2)

because the subgroups consisting of parallel translations in W̃(𝔱)_{𝔲} and W̃(𝔲) are

{e} ▷< (𝔲 ∩ Γ_{𝔱}) and {e} ▷< Γ_{𝔲},

respectively.

We will prove (5.2) by applying the wellknown argument due to I.Satake ([6] Appendix). If 𝔲 = 𝔱, then the assertion is clear. So we may assume that ρ is an outer automorphism of 𝔤. First we will show that W̃(𝔲) ⊂ W̃(𝔱)_{𝔲}. We have only to show that w_{α,n} ∈ W̃(𝔱)_{𝔲} for every α ∈ Σ(𝔲) and n by case-by-case checking. Let β be a root in Σ(𝔱) such that α = β|_{𝔲}. Note that (β, ρ(β)) ≤ 0 if ρ(β) ≠ β because β − ρ(β) ∉ Σ(𝔱) by Lemma 5.1. Let c be the least positive integer such that

ρ^{c} ∈ Int (𝔤).

Then c = 2 or 3 since 𝔤 is simple (c.f. (2.1)). Extend 𝔲 to a maximal abelian subalgebra 𝔱′ of 𝔤^{ρ^{c}}. Then 𝔱′ is maximal abelian in 𝔤 since ρ^{c} is an inner automorphism. Hence 𝔱′ = 𝔱. Thus we have proved that

ρ^{c}|_{𝔱} = id.
First suppose that c = 2. Then we have |β|/|α| = 1, \sqrt{2} or 2.
If |β|/|α| = 1, then w_{α,n} = w_{β,n}|_{𝔲} ∈ W̃(𝔱)_{𝔲}.

If |β|/|α| = \sqrt{2}, then w_{α,n} = w_{β,n}w_{ρ(β),n}|_{𝔲} ∈ W̃(𝔱)_{𝔲} since (β, ρ(β)) = 0.

If |β|/|α| = 2, then γ = β + ρ(β) ∈ Σ(𝔱) and therefore w_{α,n} = w_{γ,2n}|_{𝔲} ∈ W̃(𝔱)_{𝔲}.

Next consider the case that c = 3. If ρ(β) = β, then w_{α,n} = w_{β,n}|_{𝔲} ∈ W̃(𝔱)_{𝔲}. On the other hand, if ρ(β) ≠ β, then ∠βOρ(β) = π/2, 2π/3 or π since (β, ρ(β)) ≤ 0.

If ∠βOρ(β) = π, then ρ(β) = −β. Since ρ^{3}|_{𝔱} = id., we have β = ρ^{3}(β) = −β, a contradiction.

If ∠βOρ(β) = 2π/3, then we have

∠ρ(β)Oρ^{2}(β) = ∠ρ^{2}(β)Oβ = \frac{2π}{3} .

Hence we have

β + ρ(β) + ρ^{2}(β) = 0

which implies α = β|_{𝔲} = (1/3)(β + ρ(β) + ρ^{2}(β))|_{𝔲} = 0, a contradiction.

So we have ∠βOρ(β) = ∠ρ(β)Oρ^{2}(β) = ∠ρ^{2}(β)Oβ = π/2. Then it is clear that

w_{α,n} = w_{β,n}w_{ρ(β),n}w_{ρ^{2}(β),n}|_{𝔲} ∈ W̃(𝔱)_{𝔲}.

Conversely, we will show that W̃(𝔱)_{𝔲} ⊂ W̃(𝔲). Let w be an element of W̃(𝔱)^{𝔲} = {w ∈ W̃(𝔱) | w(𝔲) = 𝔲} and Δ_{𝔲} = Δ_{𝔱} ∩ 𝔲 an element of C(𝔲) (Δ_{𝔱} ∈ C(𝔱)). Then wΔ_{𝔲} = wΔ_{𝔱} ∩ 𝔲 ∈ C(𝔲) and there exists a w_{0} ∈ W̃(𝔲) such that w_{0}^{−1}wΔ_{𝔲} = Δ_{𝔲}. Since W̃(𝔲) ⊂ W̃(𝔱)_{𝔲}, we can choose w̃_{0} ∈ W̃(𝔱)^{𝔲} such that w̃_{0}|_{𝔲} = w_{0}. Then we have w̃_{0}^{−1}wΔ_{𝔲} = Δ_{𝔲} and therefore w̃_{0}^{−1}wΔ_{𝔱} = Δ_{𝔱}. This implies w = w̃_{0}. Hence

w|_{𝔲} = w̃_{0}|_{𝔲} = w_{0} ∈ W̃(𝔲).

Let 𝔤, σ, τ and 𝔞 be as in Section 1. Let 𝔲 be a maximal abelian subalgebra of 𝔤^{στ} containing 𝔞. Then Γ_{𝔞} is also defined by (5.1) for ∗ = 𝔞.

Proposition 5.3. Γ_{𝔞} = Γ_{𝔲} ∩ 𝔞.

Proof. As in the proof of Proposition 5.2, we can also define

W̃(𝔞) = W (𝔞) ▷< Γ_{𝔞}

and C(𝔞) in the same way. Note that for every Δ_{𝔞} ∈ C(𝔞), we can write

\overline{Δ_{𝔞}} = \overline{Δ_{𝔲}} ∩ 𝔞

with some Δ_{𝔲} ∈ C(𝔲).

Put W̃(𝔲)_{𝔞} = {w ∈ W̃(𝔲) | w(𝔞) = 𝔞}|_{𝔞}. Then we have only to show that

W̃(𝔲)_{𝔞} = W̃(𝔞)

as in the proof of Proposition 5.2.

The following lemma will be proved later.

Lemma 5.4. Let α be a root in Σ(𝔲) such that σ(α) ≠ −α. Then (α, σ(α)) ≥ 0.

Let α ∈ Σ(𝔞) and let β be a root in Σ(𝔲) such that β|_{𝔞} = α. Then by Lemma 5.4, we have |β|/|α| = 1, \sqrt{2} or 2. Hence we can prove W̃(𝔞) ⊂ W̃(𝔲)_{𝔞} by the same argument as in the proof of (5.2).

Next we will show that W̃(𝔲)_{𝔞} ⊂ W̃(𝔞). Let w be an element of W̃(𝔲)^{𝔞} = {w ∈ W̃(𝔲) | w(𝔞) = 𝔞}. Let Δ_{𝔞} be an element of C(𝔞) and Δ_{𝔲} an element of C(𝔲) such that

\overline{Δ_{𝔞}} = \overline{Δ_{𝔲}} ∩ 𝔞.

Then we have w\overline{Δ_{𝔞}} = w\overline{Δ_{𝔲}}∩𝔞 and there exists a w_{0} ∈ W̃(𝔞) such that w_{0}^{−1}wΔ_{𝔞} = Δ_{𝔞}. Since W̃(𝔞) ⊂ W̃(𝔲)_{𝔞}, we can choose w̃_{0} ∈ W̃(𝔲)^{𝔞} such that w̃_{0}|_{𝔞} = w_{0}. Let Y be an element of Δ_{𝔞}. Then we have

w̃_{0}^{−1}w(Y) = Y

since \overline{Δ_{𝔲}} is a complete set of representatives of W̃(𝔲)-orbits on 𝔲. Hence we have w̃_{0}^{−1}w|_{𝔞} = id. and therefore

w|_{𝔞} = w̃_{0}|_{𝔞} = w_{0} ∈ W̃(𝔞).

Proof of Lemma 5.4. Let α be a root in Σ(𝔲) such that σ(α) ≠ −α and X_{α} a nonzero element of 𝔤_{ℂ}(𝔲, α) such that

στX_{α} = λX_{α}

with some λ ∈ U(1). Suppose that (α, σ(α)) < 0. Then

X = [X_{α}, σX_{α}]

is a nonzero element of 𝔤_{ℂ}(𝔲, β) where β = α + σ(α) ≠ 0. We have

σX = [σX_{α}, X_{α}] = −[X_{α}, σX_{α}] = −X

and

στ(σX_{α}) = σ(στ)^{−1}X_{α} = λ^{−1}σX_{α}.

Hence we have

X ∈ 𝔤_{ℂ}(𝔲, β) ∩ 𝔤_{ℂ}^{−σ} ∩ 𝔤_{ℂ}^{στ} = 𝔤_{ℂ}(𝔲, β) ∩ 𝔤_{ℂ}^{−σ} ∩ 𝔤_{ℂ}^{−τ}.

On the other hand, we have [X, 𝔞] = {0} since β|_{𝔞} = 0. Thus we have a contradiction to the assumption that 𝔞 is maximal abelian in 𝔤^{−σ} ∩ 𝔤^{−τ}.

As a corollary of Proposition 5.2 and Proposition 5.3, we have:

Corollary . If G is simply connected, then

{Y ∈ 𝔞 | exp Y = e} = Γ_{𝔞}.

References

[1] Berger, M., Les espace symétriques non compacts, Ann. Sci. École Norm. Sup. 74 (1957), 85–177.

[2] Borel, A., Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tôhoku Math. J. 13 (1961), 216–240.

[3]  Loos, O., Symmetric spaces II, Benjamin, New York, 1969.

[4] Matsuki, T., Double coset decompositions of reductive Lie groups arising from two involutions., Journal of Algebra 197 (1997), 49–91.

[5] Oshima, T., and J. Sekiguchi, The restricted root system of a semisimple symmetric pair, Advanced Studies in Pure Math. 4 (1984), 433–497.

[6] Satake, I., On representations and compactifications of a symmetric Riemannian spaces, Ann. of Math. 71 (1960), 77–110.

[7] de Siebenthal, J., Sur les groupes de Lie compacts non connexes, Comm. Math. Helv 31 (1956), 41–89.

Toshihiko Matsuki Faculty of Integrated Human Studies Kyoto University Kyoto 606-8501, Japan matsuki@math.h.kyoto-u.ac.jp

Received September 5, 2000 and in final form April 2, 2001