Journal of Lie Theory Volume 5 (1996) 275–291 ◯^{C} 1996 Heldermann Verlag

On the Exponential Map of Almost Simple Real Algebraic Groups ^{∗}

D. Ž. D – oković ^{†} and Nguyêñ Q. Thǎńg

Communicated by K. H. Hofmann

Abstract. One of the basic problems for Lie groups is to describe the image of the exponential map. A Lie group H is called exponential resp. weakly exponential if its exponential map is surjective resp. has dense image in H. In this paper we classify all exponential groups which are identity components of almost simple real algebraic groups. The weakly exponential groups of this type were determined in our previous paper [15].

0. Introduction

The well-known notion of exponential map in the theory of Lie groups and related topics is as old as the theory itself. As one of key notions, it plays an important role in the theory as well as in other branches of mathematics, e.g. differential geometry, (transcedental) number theory, arithmetic algebraic geometry, functional analysis, dynamical systems, mathematical physics, etc. (see e.g. [2, 3, 4, 8, 22, 25]). However, until recently, the problem of describing the image of the exponential map, one of the basic and difficult problems in the theory, is still open, apart from partial results for some particular Lie groups (see e.g. [9, 10, 11, 12, 17, 19, 20, 21, 23, 24]). This problem gives rise to the following basic questions :

A) Characterize the Lie groups for which the exponential map has dense image, and

B) Characterize the Lie groups, for which the exponential map is surjective.

The Lie groups satisfying the condition mentioned in A (resp. B) are called weakly exponential (w.e.) (after [17, 18], where the question A was also raised) (resp. exponential).

ISSN 0940–2268 / $2.50 ◯^{C} Heldermann Verlag

∗ 1991 AMS Classification : Primary 20G; Secondary 22E, 11E

† Supported in part by the NSERC Grant A-5285.

Dixmier has carried out in [9] a detailed investigation of the image of the exponential map for simply connected solvable Lie groups. Therefore the problem is focused on reductive Lie groups. One of the important classes of connected reductive Lie groups, which are frequently used in practice, is the class Σ consisting of real Lie groups G(R)^{∘}, where G is an almost simple algebraic group defined over R, G(R) the group of real points of G, and ^{∘} denotes the identity component.

In our recent paper [15] we have obtained a classification of groups in Σ which are w.e. (a solution of Question A for Σ). Here, using this result we give a solution of Question B for the class Σ.

It may be of interest to mention that among all almost simple (connected) complex Lie groups only the groups PSL_{n}(C) are exponential, see [19, 20, 21].

For connected real semisimple groups H, there is a simple criterion, due to Borel ([18, Theorem 2.10]), which can be used to decide whether H is w.e. or not. We have found an analogous criterion for exponentiality of groups belonging to the class Σ. In order to apply this criterion we need to know the structure of the centralizers of unipotent elements. In most cases the necessary facts can be found in the literature. In the case of the classical group SO^{∗}(2n), the group which preserves a non-degenerate skew-hermitian form on a finite dimensional quaternionic vector space, the structure of these centralizers is not available in the sources that we have consulted. We have described these centralizers in detail in Section 5.

1. Statement of the main result

By Z_{k} we denote a cyclic group of order k, by R the real numbers, by C the complex numbers, and by H the quaternion algebra over R with standard involution. For a Lie group G with Lie algebra 𝔤 we denote by E_{G} the image of the exponential map exp : 𝔤 → G. We use also standard notations for classical Lie groups. In particular, by Spin^{∗}(2n) we denote the double cover of SO^{∗}(2n), which is also the group of real points of Spin_{2n}(C) for appropriate real structure. If W is a non-degenerate subspace of V , we denote by Spin^{∗}(W) the inverse image of SO^{∗}(2n) in Spin^{∗}(2n). If dim W = k, then Spin^{∗}(W) ≃ Spin^{∗}(2k). If k = 1, then SO^{∗}(W) is the circle group SO(2), and the same is true for Spin^{∗}(W). If k = 2, then Spin^{∗}(W) ≃ SU(2) × SU(1, 1).

In our previous paper [15] we have obtained the list of all w.e. groups in Σ. Our main objective in this paper is to determine which of these groups are in fact exponential. It is well-known [16] that all compact groups from Σ are exponential. Moreover, if G → H is a surjective homomorphism of connected Lie groups with discrete kernel (we write G ≥ H) and G is (weakly) exponential, then H is also (weakly) exponential. Thus we need only to describe non-compact exponential groups which are ”maximal” with respect to ≥.

For convenience of the reader we list in Table 1 all non-compact almost simple w.e. groups in Σ which are maximal with respect to the above partial order.

Table 1

Maximal non-compact almost simple w.e. groups in Σ

A I  PSL_{2}(R)
A II  SL_{n}(H), n ≥ 2
A III  SU(p, q), p > q ≥ 1
SU(p, p)/Z_{2^{m}}, 2p = 2^{m}r, r odd
B I  Spin(2n, 1), n ≥ 2
C II  Sp(p, q), p ≥ q ≥ 1
Spin(2n − 1, 1), n ≥ 3
D I  PSO(2n − 2, 2)^{0}, n ≥ 3, n odd
Spin(2n − 2, 2)/⟨z⟩, n ≥ 4, n even
Spin^{∗}(2n), n ≥ 3, n odd
D III  SO^{∗}(2n), n ≥ 4, n even
Spin^{∗}(2n)/⟨z′⟩, n ≥ 4, n even
E III, IV  G^{∗}
E VII  G^{∗}/Z_{2}
F II  G^{∗}

In this table z and z′ are elements of order two which generate the center Z ≃ Z_{2} ×Z_{2} of Spin_{2n}(C), such that ⟨zz′⟩ is the kernel of the canonical projection Spin_{2n}(C) → SO_{2n}(C). It is well-known that in both cases, D I and D III, Z is contained in the group of real points Spin(2n − 2, 2) and Spin^{∗}(2n), respectively. Moreover there is an automorphism of Spin(2n − 2, 2) which interchanges z and z′, but there is no such automorphism of Spin^{∗}(2n) if n is even. In the latter case we can distinguish z and z′ by considering any compact torus T of dimension n/2 which is centralized by a maximal split torus of Spin^{∗}(2n). Then z ∈ T and z′ ∉ T (see [15] for details).

We can now state our main result.

Main Theorem. The non-compact exponential groups in Σ are the groups listed below and their quotients by finite central subgroups :

1. SL_{n}(H), n ≥ 2 ;
2. PSU(p, p) = SU(p, p)/Z_{2p}, p ≥ 1 ;

3. SU(p, q)/Z_{d}, where p > q ≥ 1, d is an odd divisor of n = p + q, and every odd prime dividing d resp. n/d is ≤ n/(p − q) resp. > n/(p − q) ; 4. Spin(2n, 1), n ≥ 2 ;

5. Sp(p, q), p ≥ q ≥ 1 ;

6. Spin(2n − 1, 1), n ≥ 3 ;

7. PSO(2n − 2, 2)^{0}, n odd ≥ 3 ;

8. Spin(2n − 2, 2)/⟨z⟩, n even ≥ 4 ;

9. Spin^{∗}(2n), n odd ≥ 3 ;

10. SO^{∗}(2n), n even ≥ 4 ;

11. Spin^{∗}(2n)/⟨z′⟩, n even ≥ 4 ;

12. G^{∗} of type E IV.

In the last three rows of Table 1 we denote by G^{∗} the group of real points of the simply connected algebraic group of indicated type. It is known that PSL_{2}(R) ≃ SU(1, 1)/Z_{2} is exponential and so are the groups listed in the cases 1, 5, and 10, see [10]. Hence we can omit from further considerations the groups of type A I, A II, and C II listed in Table 1. For all other types, listed in Table 1, the proof will be given separately in the remaining sections.

2. A criterion for exponentiality

In this section we give a general criterion for exponentiality of groups belonging to Σ. Theorem 2.2 below plays an important role in our study, just like Borel’s theorem does in the study of weak exponentiality of Lie groups, see [15], [18, Theorem 2.10]. It is an analog, in the real case, of [12, Theorem 3.2]. Recall that a Cartan subgroup of a connected reductive real Lie group G is the centralizer in G of a Cartan subalgebra of the Lie algebra of G, see [28, p. 108]. First we need the following mild extension of Borel’s theorem.

Theorem 2.1. A connected reductive real Lie group G is w.e. if and only if all Cartan subgroups of G are connected.

Proof. If G is w.e. the same holds true for G/R, where R is the radical of G. If C is a Cartan subgroup of G then C ⊃ R and C/R is a Cartan subgroup of G/R. By Borel’s theorem [18, Theorem 2.10], C/R is connected. As R is connected, C is also connected.

Conversely, assume that all Cartan subgroups of G are connected. Since Cartan subgroups of G/R are images of the Cartan subgroups of G, they are also connected. By Borel’s theorem G/R is w.e. By [18, Corollary 2.1A, p. 273] G is w.e.

In the sequel 1-PSG means ”one-parameter subgroup”.

Theorem 2.2. Let G be the identity component of real points of an R-group H. Then G is exponential if and only if Z_{G}(u) is w.e. for every unipotent element u ∈ G.

Proof. Assume that G is exponential and let u ∈ G be unipotent. We have a Levi decomposition Z := Z_{G}(u) = RC, where R is the unipotent radical of Z and C is reductive (not necessarily connected). Let s ∈ C be semisimple and set x = su. Since x ∈ E_{G}, x ∈ P where P is a 1-PSG of Z_{G}(x). Since Z_{G}(x) ⊂ Z and u is unipotent, it follows that \overline{P ⟨u⟩} is a connected abelian Lie subgroup of Z containing s = xu^{−1}. Hence s ∈ E_{Z}, and C ≃ Z/R implies that s ∈ E_{C}. It is known [28, Section 1.4] that semisimple elements are dense in C, and so C is w.e. By using the fact that R is exponential and [18, Corollary 2.1A], we conclude that Z is w.e.

We now prove the converse. Let x ∈ G and let x = su be its Jordan decomposition in H(R), with s semisimple and u unipotent. Then necessarily u ∈ G, and so s ∈ Z := Z_{G}(u). We can choose a Levi decomposition Z = RC such that s ∈ C. By hypothesis Z is w.e. This entails that C ≃ Z/R is w.e. As s is semisimple, it belongs to a Cartan subgroup P of C, which is connected by Theorem 2.1. Hence x = su belongs to the connected abelian Lie group \overline{P ⟨u⟩} and so x ∈ E_{G}.

We need the following lemmas in the sequel.

Lemma 2.3. Let G = G_{1} × ⋯ × G_{n} where G_{i} are connected Lie groups, and let z_{i} ∈ G_{i} be central elements of order 2. Assume that G_{1} and G_{i}/⟨z_{i}⟩, i > 1, are exponential. Then Ḡ := G/⟨z_{1}z_{2}, z_{1}z_{3}, …, z_{1}z_{n}⟩ is exponential.

Proof. Let x = x_{1} ⋯ x_{n} ∈ G, with x_{i} ∈ G_{i}, and let x̄ ∈ Ḡ be its image. Since each of the groups G_{i}/⟨z_{i}⟩ is exponential, we can choose k_{i} ∈ {0, 1}, i > 1, such that x_{i}z_{i}^{k_{i}} ∈ E_{G_{i}}. If k_{1} = k_{2} + ⋯ + k_{n} and z = z_{1}^{k_{1}} ⋯ z_{n}^{k_{n}} then xz ∈ E_{G} because G_{1} is exponential. As z̄ = 1, we have x̄ ∈ EḠ.

The group Ḡ in the above lemma will be denoted by ∏^{∗} G_{i} and we refer to it as a direct product of G_{i}’s with identified subgroups ⟨z_{i}⟩.

Lemma 2.4. If ϕ : G → H is a homomorphism of groups and G is a k-group, where k is a field, and X ⊂ G is such that Ker ϕ ⊂ X ∩ G(k), then

ϕ(X ∩ G(k)) = ϕ(G(k)) ∩ ϕ(X).

Proof. Let y = ϕ(g) = ϕ(x), where g ∈ G(k), x ∈ X. Then x = gz, where z ∈ Ker ϕ ⊂ G(k). Hence x ∈ X ∩ G(k), and y ∈ ϕ(X ∩ G(k)).

3. Groups of type A III

In this section we investigate the exponentiality of groups of type A III listed in Table 1. In the proof of the following proposition, and also in subsequent sections, we shall use results and notations of [10]. The readers are advised to consult this paper before reading further.

Proposition 3.1. Let G = SU(p, q)/Z_{d}, where d | n = p + q, n/d = 2^{s}t, t odd, and p ≥ q ≥ 1. Then G is exponential if and only if one of the following holds :

(a) p = q and d = n,

(b) p > q and every prime divisor of t is bigger than n/(p − q). In particular, if n is a power of 2, then G is always exponential.

Proof. Let Γ be a conjugacy class of G^{∗} = SU(p, q) and Γ̂ its image under the projection π : G^{∗} → G. Then

π^{−1}(Γ̂) = Γ ∪ θΓ ∪ ⋯ ∪ θ^{d−1}Γ,

where θ = e^{2πi/d} is a generator of Z_{d}.

So Γ̂ ⊂ E_{G} if and only if at least one of the conjugacy classes θ^{s}Γ is contained in E_{G^{∗}}. As in [10, p. 80] we may write

Γ = ∑_{i=1}^{r} Γ_{m_{i}−1}(λ_{i}, λ̄_{i}^{−1}) + ∑_{i=r+1}^{k} Γ_{m_{i}−1}^{±}(λ_{i}),  (3.1)

where |λ_{i}| > 1 (resp. |λ_{i}| = 1) for i ≤ r (resp. i > r). Let p_{Γ} be the greatest common divisor of the integers 2m_{1}, ... , 2m_{r}, m_{r+1}, ... , m_{k}. The integer q_{Γ} is defined by :

2 ∑_{i=1}^{r} m_{i} arg λ_{i} + ∑_{i=r+1}^{k} m_{i} arg λ_{i} = 2πq_{Γ},

uniquely mod p_{Γ} . We claim that G is exponential if and only if GCD(p_{Γ}, n/d) = 1

for all Γ.

For, we have the following basic relations :

2(m_{1} + ⋯ + m_{r}) + m_{r+1} + ⋯ + m_{k} = n,
(m_{1} + ⋯ + m_{r}) + [m_{r+1}/2] + ⋯ + [m_{k}/2] ≤ q.

We know that Γ ⊂ E_{G^{∗}} if and only if p_{Γ} | q_{Γ} (loc. cit.). Since p_{θ^{s}Γ} = p_{Γ} and q_{θ^{s}Γ} = q_{Γ} +ns/d, it follows that θ^{s}Γ ⊂ E_{G^{∗}} if and only if p_{Γ} | q_{Γ} +ns/d. Thus

Γ̂ ⊂ E_{G} ⟺ ∃s ∈ Z, p_{Γ} | q_{Γ} + ns/d.  (3.2)

Hence if GCD(p_{Γ}, n/d) = 1 for all Γ then G is exponential.

Conversely, assume that G is exponential and let Γ be as in (3.1). Let Γ′ ⊂ G^{∗} be obtained from Γ by replacing each λ_{i} by some λ′_{i}. As p_{Γ′} = p_{Γ} and Γ̂′ ⊂ E_{G}, (3.2) implies that p_{Γ} | q_{Γ′} + ns/d for some s ∈ Z. Since q_{Γ′} can take arbitrary integer values for appropriate choice of the λ′_{i}, it follows that GCD(p_{Γ}, n/d) = 1. Hence our claim is proved.

Assume that p = q. We can choose Γ such that p_{Γ} = 2 if n/d is even, and p_{Γ} = 2n/d if n/d is odd. Hence if d ≠ n then G is not exponential.

¿From now on we assume that p > q, which implies that p_{Γ} is odd for all Γ, see [7]. Hence if t = 1 then GCD(p_{Γ}, n/d) = 1 and G is exponential. It remains to consider the case t > 1. Assume p−q ≤ n/p_{0}, where p_{0} is the smallest prime divisor of t. Then u = (p − q + n/p_{0})/2 and v = n/p_{0} − u are nonnegative integers. For

Γ = uΓ_{p_{0}−1}^{+}(1) + vΓ_{p_{0}−1}^{−}(1),

we have p_{Γ} = p_{0} and GCD(n/d, p_{Γ}) = p_{0} > 1. Hence G is not exponential.

Conversely assume that G is not exponential. Then δ := GCD(n/d, p_{Γ}) > 1 for some Γ as in (3.1). We may assume that the m_{i} are odd for r < i ≤ r + l and even for i > r + l. From the basic relations we have

q ≥ ∑_{i=1}^{r} m_{i} + ∑_{i=r+1}^{r+l}(m_{i} − 1)/2 + ∑_{i=r+l+1}^{k} m_{i}/2 = (n − l)/2,

and so p − q = n − 2q ≤ l ≤ k ≤ kp_{Γ}/δ ≤ n/δ ≤ n/p_{0}.

The groups SU(p, q)/Z_{d} that satisfy condition (b) of this proposition are exactly the quotients of the group given in case 3 of the main theorem.

The following corollary is an immediate consequence of the proposition.

Corollary 3.2. The group SU(p, q), p ≥ q ≥ 1, is exponential if and only if p > q and every odd prime divisor of n = p + q is bigger than n/(p − q).

4. Groups of types B I and D I

In this section we investigate the exponentiality of groups of types B I and D I listed in Table 1.

Proposition 4.1. For n ≥ 5, Spin(n − 1, 1) is exponential.

Proof. Let ϕ : G^{∗} = Spin(n − 1, 1) → G = SO(n − 1, 1) be the canonical homomorphism. If u^{∗} ∈ G^{∗} is unipotent then by Theorem 2.2 we have to show that Z_{G^{∗}}(u^{∗}) is w.e. Let u = ϕ(u^{∗}) and

Z_{G}(u) = RC, Z_{G^{∗}}(u^{∗}) = R^{∗}C^{∗},

where R and R^{∗} are unipotent radicals and C and C^{∗} the Levi factors. It is well-known that R^{∗} is exponential, hence we have only to show that C^{∗} is w.e. As G^{∗} is w.e. (see Table 1), we may assume that u^{∗} ≠ 1. Then u ≠ 1 and the conjugacy class of u in O(n − 1, 1) is Γ_{2}^{+}(1) + (n − 3)Γ_{0}^{+}(1) , see [10, p. 83]. Let Z̃ be the centralizer of u in O(n − 1, 1). We have Z̃ = RC̃ where C̃ is a Levi factor. By [26, Chapter 4]), C̃ ≃ O(1) × O(n − 3), and so C̃ ∩ G^{0} ≃ SO(n − 3). By Lemma 2.4, ϕ(C^{∗}) is a Levi factor of Z_{G^{0}}(u) = Z̃ ∩ G^{0} and is isomorphic to SO(n − 3). Hence C^{∗} = ϕ^{−1}(C̃ ∩ G^{0}) ≃ Spin(n − 3) is exponential.

Proposition 4.2. For even n ≥ 4, Spin(2n − 2, 2)/⟨z⟩ is exponential.

Proof. Let ϕ : G^{∗} = Spin(2n − 2, 2) → G = SO(2n − 2, 2) and ψ : G^{∗} → G′ := G^{∗}/⟨z⟩ be the canonical maps. For a unipotent u′ ∈ G′, there is a unique unipotent u^{∗} ∈ G^{∗} such that ψ(u^{∗}) = u′, and we set u = ϕ(u^{∗}). We have Levi decompositions

Z_{G′}(u′) = R′C′, Z_{G^{∗}}(u^{∗}) = R^{∗}C^{∗}, Z_{G}(u) = RC,

where R′ ≃ R^{∗} ≃ R are unipotent, and so exponential. As above, it suffices to show that C′ is w.e. and we may assume that u′ ≠ 1. From [10, p. 83] it follows that there are exactly four non-trivial unipotent conjugacy classes Γ in O(2n − 2, 2). For each of these classes we list in Table 2 the Levi factors C̃ of the centralizer of u ∈ Γ in O(2n − 2, 2), and also C̃ ∩ G^{0}. For the description of Γ see [10, p. 83] and for C̃ see [26, Chapter 4].

Table 2 Levi factors of centralizers of unipotent elements in SO(2n − 2, 2)^{∘}

Γ  C̃  C̃ ∩ G^{0}
1 Γ_{4}^{+}(1) + (2n − 5)Γ_{0}^{+}(1) O(1) × O(2n − 5)  Z_{2} × SO(2n − 5)
2 Γ_{2}^{−}(1) + (2n − 3)Γ_{0}^{+}(1) O(1) × O(2n − 3)  Z_{2} × SO(2n − 3)
3 2Γ_{2}^{+}(1) + (2n − 6)Γ_{0}^{+}(1) O(2) × O(2n − 6) SO(2) × SO(2n − 6)
4 Γ_{1}(1, 1) + (2n − 4)Γ_{0}^{+}(1) Sp_{2}(R) × O(2n − 4) Sp_{2}(R) × SO(2n − 4)

In case 1 resp. 2 we have C^{∗} = ϕ^{−1}(C) = ⟨z⟩ × H, where H ≃ Spin(2n − 5) resp. Spin(2n − 3). Hence C′ = ψ(C^{∗}) is connected and compact. The same is true in case 3, because then C^{∗} ≃ SO(2)Spin(2n − 6).

In the last case we have C^{∗} ≃ Sp_{2}(R) × Spin(2n − 4). Since z is embedded diagonally in C^{∗}, C′ = ϕ(C^{∗}) = C^{∗}/⟨z⟩ is exponential by Lemma 2.3.

Proposition 4.3. For n ≥ 3, PSO(2n − 2, 2)^{0} is exponential.

Proof. If a ∈ SO(2n − 2, 2)^{0} then by [10, Theorem 6.3] a or −a is an exponential in this group. Consequently every element of PSO(2n − 2, 2)^{0} is an exponential.

5. Centralizers of unipotent elements in SO^{∗}(2n)

Springer and Steinberg [26, Chapter 4] have described the structure of the centralizers of unipotent elements in unitary, symplectic and orthogonal groups over any field of characteristic ≠ 2. In order to apply our exponentiality criterion, we also need a description of these centralizers for the group G = SO^{∗}(2n), consisting of all R-linear automorphisms preserving a nondegenerate skew-hermitian form on a right H-vector space V of dimension n. We shall use the same method as in [26, Chapter IV].

For an unipotent element u ∈ G, there exists a unique nilpotent element X in the Lie algebra 𝔤 of G such that u = exp(X). It is well-known that the centralizers of X and u in G coincide. We can choose a basis of V in which X has the canonical form as follows, see [7] and [11, pp. 224-225].

There are vectors e_{i}, f_{j}, g_{k} in V (i ∈ I, j ∈ J, k ∈ K) and nonnegative even (resp. odd) integers α_{i} (resp. β_{j}, γ_{k}) such that X^{α_{i}+1}e_{i} = X^{β_{j}+1}f_{j} = X^{γ_{k}+1}g_{k} = 0, and the vectors

e_{i}, Xe_{i}, X^{2}e_{i}, …, X^{α_{i}}e_{i},

f_{j}, Xf_{j}, X^{2}f_{j}, …, X^{β_{j}}f_{j},  (5.1)

g_{k}, Xg_{k}, X^{2}g_{k}, …, X^{γ_{k}}g_{k}

form a basis of V . Moreover their scalar products are

⟨X^{a}e_{i}, X^{α_{i}−a}e_{i}⟩ = (−1)^{a}\sqrt{−1}, 0 ≤ a ≤ α_{i},
⟨X^{b}f_{j}, X^{β_{j}−b}f_{j}⟩ = (−1)^{b}, 0 ≤ b ≤ β_{j},
⟨X^{c}g_{k}, X^{γ_{k}−c}g_{k}⟩ = (−1)^{c}, 0 ≤ c ≤ γ_{k},

and all other products are zero.

Let i be an even resp. odd integer and let V_{i}(0) be the R-span of

{e_{r} : α_{r} = i}

resp.

{f_{s} : β_{s} = i} ∪ {g_{t} : γ_{t} = i}.

We set also

V_{i}(s) = X^{s}V_{i}(0) for s ≤ i,

and

V_{i} = V_{i}(0) + ⋯ + V_{i}(i).

By X_{i} we denote the restriction of X to V_{i}, and by Z_{i} the centralizer of X_{i} in SO^{∗}(V_{i}). Define a one-parameter subgroup λ : R^{∗} → G by

λ(t)X^{a}e_{i} = t^{2a−α_{i}}X^{a}e_{i},
λ(t)X^{b}f_{j} = t^{2b−β_{j}}X^{b}f_{j},
λ(t)X^{c}g_{k} = t^{2c−γ_{k}}X^{c}g_{k}.

¿From above it follows immediately that V_{i} are X -invariant and V = ⊕_{i}V_{i}.

Lemma 5.1. λ(t) normalizes Z_{G}(X) and all weights of λ in the centralizer 𝔷 of X in 𝔤 are nonnegative.

Proof. We have λ(t)X = t^{2}Xλ(t) by direct computation on basis vectors. Consequently λ(t) normalizes Z_{G}(X). We prove that the weights of λ in 𝔷 are nonnegative. Assume that Y ∈ 𝔷 has weight p, i.e., Y ≠ 0 and λ(t)Y λ(t)^{−1} = t^{p}Y . Then

λ(t)Y e_{i} = t^{p}Y λ(t)e_{i} = t^{p}Y t^{−α_{i}}e_{i} = t^{p−α_{i}}Y e_{i}

and similarly λ(t)Y f_{j} = t^{p−β_{j}}Y f_{j} and λ(t)Y g_{k} = t^{p−γ_{k}}Y g_{k}. As Y ≠ 0, at least one of the vectors Y e_{i}, Y f_{j}, Y g_{k} is nonzero. Say Y e_{i} ≠ 0. Assume that X^{b}f_{j} occurs in Y e_{i}. Then X^{α_{i}+1}Y e_{i} = 0 implies that X^{α_{i}+1}X^{b}f_{j} = 0, i.e. α_{i} +b ≥ β_{j}. As Y e_{i} has weight p−α_{i}, the vector X^{b}f_{j} must have the same weight, i.e. 2b−β_{j} = p−α_{i}. Hence p = 2b + α_{i} − β_{j} ≥ b ≥ 0. The other cases are similar.

Now we come to the main result of this section.

Proposition 5.2. Let Γ ⊂ G = SO^{∗}(2n) be a unipotent conjugacy class,

u ∈ Γ = ∑_{i even} k_{i}Γ_{i}(1, 1) + ∑_{i odd}(p_{i}Γ_{i}^{+}(1, 1) + q_{i}Γ_{i}^{−}(1, 1)),

and let Z := Z_{G}(u) = RC, where R is the unipotent radical and C a Levi factor of Z. Then C = ∏_{i≥0} C_{i} where

C_{i} ≃ SO^{∗}(2k_{i}), i even; C_{i} ≃ Sp(p_{i}, q_{i}), i odd.

Proof. Let X ∈ 𝔤 be the nilpotent element such that u = exp(X). Then it is well-known that Z is also the centralizer of X in G. By Lemma 5.1, 𝔷 = 𝔷^{0} ⊕ 𝔷^{+} where 𝔷^{0} is the centralizer of λ in 𝔷, i.e., the zero weight space of λ, and 𝔷^{+} is the nilpotent ideal which is the sum of all other weight spaces. Since the centralizer of λ in Z is reductive (see [26, Section 2.23]), it follows that 𝔷^{+} is the Lie algebra of R and we may assume that C is the centralizer of λ in Z.

Let r be the largest α_{i}. Then it is not hard to see that V_{r}(0) is a weight space since all basis vectors of V are also weight vectors. Hence V_{r}(0) is also C -invariant. (Indeed, if x ∈ C and α_{i} = r then λ(t)xe_{i} = xλ(t)e_{i} = t^{−r}xe_{i}.) Therefore all the spaces X^{t}(V_{r}(0)), t ≤ r are also C -invariant. In particular, the weight space corresponding to the weight −r is the direct sum of C -invariant subspaces.

Assume that there exists α_{i} ≠ r and let s be the largest such α_{i}. Since r and s are even, r − s = 2k. By considering the basis vectors as above, one can see that the space W := V_{r}(k) + V_{s}(0) is the weight space of weight −s, and so W is C -invariant. As C is reductive and V_{r}(k) is C -invariant, there exists a C - invariant complement W_{s} of V_{r}(k) in W. By replacing V_{s}(0) by W_{s}, X^{t}V_{s}(0) by X^{t}W_{s}, t ≤ s we may assume that the weight space W corresponding to the weight −s is the direct sum of C -invariant subspaces. Meanwhile one should note that the subspace W_{s} consists of vector of the same weight −s. Thus we may change the original basis of V_{s}(0) by a new one via the change V_{s}(0) by W_{s}. Therefore we may assume that V_{s}(0) is C -invariant. By repeating this argument, we may assume that all the spaces V_{i}(0) (and therefore also the spaces X^{t}V_{i}(0), t ≤ i) are C -invariant subspaces. Consequently C = ∏ C_{i}, where C_{i} is the centralizer of λ in Z_{i}.

Now we define a sesquilinear form ϕ_{i} : V_{i}(0) × V_{i}(0) → H by
ϕ_{i}(v, w) = ⟨v, X^{i}w⟩.

Then

ϕ_{i}(w, v) = ⟨w, X^{i}v⟩ = ⟨(−X)^{i}w, v⟩
= (−1)^{i+1}\overline{⟨v, X^{i}w⟩} = (−1)^{i+1}\overline{ϕ_{i}(v, w)},

and so ϕ_{i} is skew-hermitian for even i and hermitian for odd i. It is also non-degenerate. Indeed, if i is even, we have ϕ_{i}(e_{i_{r}}, e_{i_{s}}) = 0 if r ≠ s and ϕ_{i}(e_{i_{r}}, e_{i_{r}}) = −\sqrt{−1}. If i is odd then the basis f_{j_{1}}, …, f_{j_{p}}, g_{l_{1}}, …, g_{l_{q}} of V_{i}(0) is orthogonal with respect to ϕ_{i} and ϕ_{i}(f_{j_{r}}, f_{j_{r}}) = 1, ϕ_{i}(g_{l_{s}}, g_{l_{s}}) = −1. Hence ϕ_{i} has signature (p, q).

It is clear that C_{i} preserves ϕ_{i}. Indeed for x ∈ C_{i} and v, w ∈ V_{i}(0) we have

ϕ_{i}(x(v), x(w)) = ⟨x(v), X^{i}x(w)⟩ = ⟨x(v), x(X^{i}w)⟩
= ⟨v, X^{i}w⟩ = ϕ_{i}(v, w).

We claim that C_{i} ≃ SO^{∗}(ϕ_{i}) for i even, and C_{i} ≃ U(ϕ_{i}), the unitary group of ϕ_{i}, for i odd.

For, every x in SO^{∗}(ϕ_{i}) (or U(ϕ_{i})) extends uniquely to an automorphism x̃ of V_{i} commuting with X. Say x ∈ SO^{∗}(ϕ_{i}), and let

xe_{r} = ∑_{s} e_{s}b_{s}, α_{r} = i, b_{s} ∈ H, e_{s} ∈ V_{i}(0).

Then

⟨x̃X^{a}e_{r}, x̃X^{i−a}e_{r}⟩ = ⟨X^{a}x̃e_{r}, X^{i−a}x̃e_{r}⟩
= ⟨X^{a} ∑_{s} e_{s}b_{s}, X^{i−a} ∑_{t} e_{t}b_{t}⟩
= ∑_{s,t} b̄_{s}⟨X^{a}e_{s}, X^{i−a}e_{t}⟩b_{t}
= ∑_{s} b̄_{s}⟨X^{a}e_{s}, X^{i−a}e_{s}⟩b_{s}
= (−1)^{a} ∑_{s} b̄_{s}\sqrt{−1}b_{s}
= (−1)^{a}ϕ_{i}(xe_{r}, xe_{r}) = (−1)^{a}ϕ_{i}(e_{r}, e_{r})
= (−1)^{a}⟨e_{r}, X^{i}e_{r}⟩
= (−1)^{a}\sqrt{−1}
= ⟨X^{a}e_{r}, X^{i−a}e_{r}⟩.

The same holds for the scalar products of other basis vectors. This means that this extension x̃ preserves the form ⟨., .⟩ on V_{i}, i.e., x̃ ∈ C_{i}. Hence our claim (and the proposition) is proved.

We shall need later a certain overgroup of C_{i} in SO^{∗}(V_{i}) when i is even. We shall now describe this overgroup (see (5.2) below). We know that C_{i} leaves invariant each of the subspaces V_{i}(s) and commutes with the action of X. The space V_{i} is an orthogonal direct sum of non-degenerate subspaces V_{i}(s)+V_{i}(i−s), for 0 ≤ s < i/2, and V_{i}(i/2). Hence

SO^{∗}(V_{i}(i/2)) × ∏_{0≤s<i/2} SO^{∗}(V_{i}(s) + V_{i}(i − s))

is a subgroup of SO^{∗}(V_{i}).

It is well-known that if V, W are totally isotropic spaces (with respect to some non-degenerate quadratic or (skew-) hermitian form) of the same dimension such that V + W is non-degenerate then the intersection

(GL(V) × GL(W)) ∩ SO(V + W)

is isomorphic to GL(V). In particular, since V_{i}(s) and V_{i}(i − s) are totally isotropic, the subgroup

H_{i}(s) = (GL(V_{i}(s)) × GL(V_{i}(i − s))) ∩ SO^{∗}(V_{i}(s) + V_{i}(i − s))

is isomorphic to GL(V_{i}(s)). The group C_{i} is a subgroup of

SO^{∗}(V_{i}(i/2)) × ∏_{0≤s<i/2} H_{i}(s)  (5.2)

and the projection of C_{i} in each factor of (5.2) is isomorphic to C_{i}. Thus C_{i} is ”diagonally” embedded in the above direct product.

Lemma 5.3. Let ϕ : Spin^{∗}(2n) → SO^{∗}(2n) be the canonical projection. If i is even and V_{i} ≠ 0 then ϕ^{−1}(C_{i}) ≃ Spin^{∗}(2k_{i}).

Proof. Set

V′_{i} = ∑_{0≤s<i/2} V_{i}(s), V′′_{i} = ∑_{i/2<s≤i} V_{i}(s).  (5.3)

By the above remark, the group

H_{i} = (GL(V′_{i}) × GL(V′′_{i})) ∩ SO^{∗}(V′_{i} + V′′_{i})

is isomorphic to GL(V′_{i}). Put V_{i}^{0} = V_{i}(i/2). Since C_{i} ⊂ SO^{∗}(V_{i}^{0}) × H_{i} and H_{i} is simply connected (see [16]), it follows that

ϕ^{−1}(C_{i}) ⊂ ϕ^{−1}(SO^{∗}(V_{i}^{0}) × H_{i}) = Spin^{∗}(V_{i}^{0}) × H_{i}^{∗},

where H_{i}^{∗} ≃ H_{i}. Hence ϕ^{−1}(C_{i}) ≃ Spin^{∗}(V_{i}^{0}).

6. Special tori in groups of type D III

Let G be a semisimple R-group. An anisotropic R-torus T ⊂ G is called special if Z_{G}(T) contains a maximal R-split torus S of G such that ST is a maximal R-torus of G. Since the maximal R-split tori are G(R)-conjugate (see [5, Section 5], the same is true for special tori.

Let n be even, and G̃ = Spin_{2n}(C) with G̃(R) = G^{∗} = Spin^{∗}(2n). We denote by {±1} the kernel of the canonical projection Spin_{2n}(C) → SO_{2n}(C). If T is special in G̃, then precisely one of the elements of the center of G^{∗}, distinct from {±1}, belongs to T, see [15, Proof of Theorem 4.5]. In order to conform to the notation of that paper, let Z̃ = {1, z, z′, zz′} be the center of G^{∗}, with zz′ = −1, z ∈ T, and z′ ∉ T. We also note that for any maximal R-split torus S we have S ∩ Z̃ = {1} (loc.cit.). In the next three lemmas, we refer to the real points of special (resp. split) tori also as special (resp. split) tori, which should not lead to any confusion. We shall use the notations V, V_{i}, V_{i}(s), C_{i} etc. from the previous section.

Lemma 6.1. If V = U_{1} ⊕ U_{2} and U_{1} ⊥ U_{2}, then the product of special tori of SO^{∗}(U_{1}) and SO^{∗}(U_{2}) is contained in a special torus of SO^{∗}(V).

Proof. Let T_{i} be a special torus of SO^{∗}(U_{i}), i = 1, 2. Then there is a maximal split torus S_{i} of SO^{∗}(U_{i}) centralizing T_{i}. Since S_{1}S_{2} is a maximal split torus of SO^{∗}(V) which centralizes T_{1}T_{2}, the lemma follows.

Lemma 6.2. Every special torus of C_{i}, i even, is contained in a special torus of SO^{∗}(V_{i}).

Proof. We define V′_{i} and V′′_{i} by (5.3) and set V_{i}^{0} = V_{i}(i/2). The spaces V′_{i} and V′′_{i} are totally isotropic. Their sum and V_{i}^{0} are non-degenerate. As above, the group

H_{i} = (GL(V′_{i}) × GL(V′′_{i})) ∩ SO^{∗}(V′_{i} + V′′_{i})

is isomorphic to GL(V′_{i}). By (5.2) we have

C_{i} ⊂ SO^{∗}(V_{i}^{0}) × H_{i} ⊂ SO^{∗}(V_{i}^{0}) × SO^{∗}(V′_{i} + V′′_{i}) ⊂ SO^{∗}(V_{i}).

Let P be a special torus of C_{i}. Since the projection of C_{i} to SO^{∗}(V_{i}^{0}) is an isomorphism (see (5.2)), the image P′ of P under this projection is also a special torus of SO^{∗}(V_{i}^{0}). Since P is compact, its projection P′′ to H_{i} is contained in a maximal compact torus Q of H_{i}. Since Q is special in SO^{∗}(V′_{i}+V′′_{i}), the assertion follows from Lemma 6.1.

Lemma 6.3. Let P_{i} be a torus of C_{i} such that P_{i} is special in C_{i} for i even, and P_{i} is maximal compact in C_{i} for i odd. Then the product P of all P_{i}’s is contained in a special torus of SO^{∗}(V).

Proof. By Lemma 6.1 it suffices to show that each P_{i} is contained in a special torus of SO^{∗}(V_{i}). If i is even this was proved in Lemma 6.2. If i is odd, it follows from the fact that we have the inclusions C_{i} ⊂ H_{i} ⊂ SO^{∗}(V_{i}), where H_{i} is defined analogously to the case when i is even. Indeed, P_{i} is a maximal compact torus of H_{i} and every such torus is special in SO^{∗}(V_{i}).

7. Groups of type D III

In this section we investigate the exponentiality of the groups of type D III listed in Table 1. We set G^{∗} = Spin^{∗}(2n) and G = SO^{∗}(2n). Since G is exponential, there are two cases to consider depending on the parity of n. We first deal with the case when n is odd.

Proposition 7.1. For odd n ≥ 3, Spin^{∗}(2n) is exponential.

Proof. We use induction on n. If n = 3, then Spin^{∗}(6) ≃ SU(3, 1) is exponential by Corollary 3.2. Let n > 3 and assume that the assertion is true for all odd integers < n. Let u^{∗} ∈ Spin^{∗}(2n), u^{∗} ≠ 1, be a unipotent element, and u = ϕ(u^{∗}), where ϕ : G^{∗} → G is the natural projection. The centralizer Z = Z_{G}(u) is described in Proposition 5.2. Namely, Z = RC where R is the unipotent radical and the Levi factor C = ∏_{i≥0} C_{i}. Furthermore, C_{i} ≃ SO^{∗}(2k_{i}) for i even, and C_{i} ≃ Sp(p_{i}, q_{i}) for i odd.

Since n is odd, there is at least one even integer i such that k_{i} is odd and, necessarily, k_{i} < n (for the notations see Section 5). By Lemma 5.3, ϕ^{−1}(C_{i}) ≃ Spin^{∗}(2k_{i}) and, by induction hypothesis, Spin^{∗}(2k_{i}) is exponential. Since

ϕ^{−1}( ∏_{i even} C_{i}) ≃ ∏_{i even}^{∗} Spin^{∗}(2k_{i}),

this group is exponential by Lemma 2.3.

If i is odd then it is well-known that C_{i} is simply connected, and so ϕ^{−1}(C_{i}) = {±1} × C_{i}^{∗} with C_{i}^{∗} ≃ C_{i}. By Lemma 5.3, for i even, C_{i}^{∗} := ϕ^{−1}(C_{i}) ≃ Spin^{∗}(2k_{i}). Consequently we have

C^{∗} = ϕ^{−1}(C) = ∏_{i even}^{∗} C_{i}^{∗} × ∏_{i odd} C_{i}^{∗}.  (7.1)

As each of the groups C_{i}, i odd, is exponential, it follows from (7.1), that the group C^{∗} = ϕ^{−1}(C) is exponential. Hence G^{∗} is exponential by Theorem 2.2.

Now we consider the case n is even.

Proposition 7.2. For even n ≥ 4, G′ := Spin^{∗}(2n)/⟨z′⟩ is exponential.

Proof. Let ϕ : G^{∗} → G and ψ : G^{∗} → G′ be the canonical projections. As we noted before, z′ does not belong to any maximal R-split torus of G^{∗}. It follows from [6, Corollary 4.7] that G′ is the group of real points of the corresponding R-group Spin_{2n}(C)/⟨z′⟩. Given a unipotent element u′ ∈ G′ there exists a unique unipotent element u^{∗} ∈ G^{∗} such that u′ = ψ(u^{∗}). Let Z_{G′}(u′) = R′C′, where R′ is the unipotent radical and C′ a Levi factor. Then Z_{G^{∗}}(u^{∗}) = R^{∗}C^{∗} where R^{∗} ≃ R′ and C^{∗} = ψ^{−1}(C′). We shall prove that C′ is w.e.

If u = ϕ(u^{∗}) then Z_{G}(u) = RC where R = ϕ(R^{∗}) and C = ϕ(C^{∗}). The Levi factor C is described in Proposition 5.2. We shall use the notations introduced in that section.

Consider first the case where C_{i} = 1 for all even i. Then C is simply connected and so C^{∗} = {±1} × C^{∗0} with C^{∗0} ≃ C.

We claim that z′ ∉ C^{∗0}. To prove this claim, we introduce the subspaces

V′ = ∑_{i odd} ∑_{0≤s<i/2} V_{i}(s),

and V′′ defined similarly except that i/2 < s ≤ i. They are maximal totally isotropic subspaces and V = V′ ⊕ V′′. We have C ⊂ H ⊂ G, where H := (((GL(V′) × GL(V′′)) ∩ G) ≃ GL(V′). By taking inverse images of the triple C ⊂ H ⊂ G under ϕ and by using the fact that H is simply connected, we obtain C^{∗} ⊂ {±1} × H^{∗} ⊂ G^{∗}, with H^{∗} ≃ H. Since maximal compact tori of H^{∗} are special in G^{∗}, we have z′ ∉ H^{∗}, and so z′ ∉ C^{∗0}. This proves our claim.

Hence C′ = ψ(C^{∗}) ≃ C and so C′ is exponential.

Now assume that C_{i} ≠ 1 for at least one even integer i. ¿From (7.1) it follows that if at least one of the k_{i}’s is odd, the corresponding C_{i}^{∗} ≃ Spin^{∗}(2k_{i}) is exponential by Prop. 7.1, and by Lemma 2.3, ϕ^{−1}(C) and C′ are exponential. It remains to consider the case where all k_{i}’s are even.

By Lemma 2.4 the maximal tori of C′ are of the form ψ(P) where P is a maximal torus of C^{∗}. Assume that P is not connected. As G is exponential, ϕ(P) is connected. Thus [P : P^{0}] = 2 and −1 ∉ P^{0}. For i even each P_{i} := P ∩C_{i}^{∗} is a maximal torus of C_{i}^{∗} and −1 ∈ P_{i} \ P_{i}^{0}. ¿From the known classification of maximal tori in C_{i} (see [15]), it follows that P_{i} is a maximally split maximal torus of C_{i}^{∗}. Consequently the maximal compact subtorus Q_{i} of P_{i} is special in C_{i}^{∗} for i even. For odd i let Q_{i} be any compact maximal torus of C_{i}^{∗}. By Lemma 6.3 the product Q of all Q_{i}’s is contained in a special torus of G^{∗}.

For i even let z_{i}, z′_{i} be the central elements of C_{i}^{∗} defined analogously to z and z′. For i odd let z_{i} be the non-identity central element of C_{i}^{∗} ≃ Sp(p_{i}, q_{i}). Let w be the product of all z_{i}. Then ϕ(w) = −1 and so w ∈ {z, z′}. Since each k_{i} is even, we have z_{i} ∈ Q_{i} for i even, see [15, Proof of Theorem 4.5], and so w ∈ Q. Since Q is contained in a special torus of G^{∗}, we must have w = z. Hence z ∈ Q ⊂ P^{0}. As P has only two components, it follows that ψ(P) = P/⟨z′⟩ is connected. This completes the proof of the proposition.

8. Exceptional groups

In this short section we investigate the exponentiality of the four types of exceptional groups listed in Table 1.

Proposition 8.1. Among the exceptional w.e. groups in Σ only the one of type E IV is exponential.

Proof. Let 𝔤 be the Lie algebra of G(R). We claim that for adjoint groups of type E III, E VII, or F II there is a nilpotent element X ∈ 𝔤 such that Z_{G(R)}(X) is not connected (as a Lie group). Then, by Theorem 2.2, G(R) is not w.e.

This claim follows by inspection of tables in [13, 14], which classify nilpotent orbits in 𝔤 under the action of the adjoint group, and the description of Z_{G}(X) for a nilpotent element X ∈ 𝔤 given in [1], see also [12, Section 5].

If G is of type E III then we choose X to lie in the orbit No. 6 in Table X of [13]. When G is of type E VII then we use orbit No. 2 in Table VII of [14]. If G is of type F II there are only two non-zero nilpotent orbits in 𝔤, see [13, Table VIII], and in both cases Z_{G(R)}(X) is disconnected.

Now let G^{∗} be of type E IV. Since the center of G^{∗} is trivial (see [T]), we may assume that G^{∗} is the group of real points of the adjoint R-group of type E IV. There are only two non-zero nilpotent orbits in 𝔤, see [14, p. 204]. The Levi factor of Z_{G^{∗}}(X) is compact and connected for all X ∈ 𝔤, X ≠ 0. By Theorem 2.2, G^{∗} is exponential.

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Department of Pure Mathematics University of Waterloo Waterloo, Ontario Canada N2L 3G1 dragomir@herod.uwaterloo.ca Department of Math. and Stat. McMaster University Hamilton, Ontario Canada L8S 4K1 nguyen@icarus.math.mcmaster.ca

Received January 10, 1996