Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 2, 451-461.

Homogeneous Lorentz Manifolds with Simple Isometry Group

Dave Witte

Department of Mathematics, Oklahoma State University Stillwater, OK 74078 e-mail: dwitte@math.okstate.edu

Abstract. Let H be a closed, noncompact subgroup of a simple Lie group G, such that G/H admits an invariant Lorentz metric. We show that if G = SO(2, n), with n ≥ 3, then the identity component H^{∘} of H is conjugate to SO(1, n)^{∘}. Also, if G = SO(1, n), with n ≥ 3, then H^{∘} is conjugate to SO(1, n − 1)^{∘}.

1. Introduction

Definition 1.1.

• A Minkowski form on a real vector space V is a nondegenerate quadratic form that is isometric to the form −x_{1}^{2} + x_{2}^{2} + ⋯ + x_{n+1}^{2} on ℝ^{n+1}, where dim V = n + 1 ≥ 2.

• A Lorentz metric on a smooth manifold M is a choice of Minkowski metric on the tangent space T_{p}M, for each p ∈ M, such that the form varies smoothly as p varies.

A. Zeghib [14] classified the compact homogeneous spaces that admit an invariant Lorentz metric. In this note, we remove the assumption of compactness, but add the restriction that the transitive group G is almost simple. Our starting point is a special case of a theorem of N. Kowalsky.

Theorem 1.2. (N. Kowalsky, cf. [11, Thm. 5.1]) Let G/H be a nontrivial homogeneous space of a connected, almost simple Lie group G with finite center. If there is a G-invariant Lorentz metric on G/H, then either

1) there is also a G-invariant Riemannian metric on G/H; or

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

2) G is locally isomorphic to either SO(1, n) or SO(2, n), for some n.

As explained in the following elementary proposition, it is easy to characterize the homogeneous spaces that arise in Conclusion (1) of Theorem 1.2, although it is probably not reasonable to expect a complete classification.

Notation 1.3. We use 𝔤 to denote the Lie algebra of a Lie group G, and 𝔥 ⊂ 𝔤 to denote the Lie algebra of a Lie subgroup H of G.

Proposition 1.4. (cf. [11, Thm. 1.1]) Let G/H be a homogeneous space of a Lie group G, such that 𝔤 is simple and dim G/H ≥ 2. The following are equivalent.

1) The homogeneous space G/H admits both a G-invariant Riemannian metric and a G-invariant Lorentz metric.

2) The closure of Ad_{G} H is compact, and leaves invariant a one-dimensional subspace of 𝔤 that is not contained in 𝔥.

The two main results of this note examine the cases that arise in Conclusion (2) of Theorem 1.2. It is well known [10, Egs. 2 and 3] that SO(1, n)^{∘}/ SO(1, n − 1)^{∘} and SO(2, n)^{∘}/ SO(1, n)^{∘} have invariant Lorentz metrics. Also, for any discrete subgroup Γ of SO(1, 2), the Killing form provides an invariant Lorentz metric on SO(1, 2)^{∘}/Γ. We show that these are essentially the only examples.

Note that SO(1, 1) and SO(2, 2) fail to be almost simple. Thus, in 1.2(2), we may assume • G is locally isomorphic to SO(1, n), and n ≥ 2; or

• G is locally isomorphic to SO(2, n), and n ≥ 3.

Proposition 2.4′. Let G be a Lie group that is locally isomorphic to SO(1, n), with n ≥ 2. If H is a closed subgroup of G, such that

• the closure of Ad_{G} H is not compact, and

• there is a G-invariant Lorentz metric on G/H, then either

1) after any identification of 𝔤 with 𝔰𝔬(1, n), the subalgebra 𝔥 is conjugate to a standard copy of 𝔰𝔬(1, n − 1) in 𝔰𝔬(1, n), or

2) n = 2 and H is discrete.

Theorem 3.5′. Let G be a Lie group that is locally isomorphic to SO(2, n), with n ≥ 3. If H is a closed subgroup of G, such that

• the closure of Ad_{G} H is not compact, and

• there is a G-invariant Lorentz metric on G/H, then, after any identification of 𝔤 with 𝔰𝔬(2, n), the subalgebra 𝔥 is conjugate to a standard copy of 𝔰𝔬(1, n) in 𝔰𝔬(2, n).

N. Kowalsky announced a much more general result than Theorem 3.5′ in [10, Thm. 4], but it seems that she did not publish a proof before her premature death. She announced a version of Proposition 2.4′ (with much more general hypotheses and a somewhat weaker conclusion) in [10, Thm. 3], and a proof appears in her Ph.D. thesis [9, Cor. 6.2].

Remark 1.5. It is easy to see that there is a G-invariant Lorentz metric on G/H if and only if there is an (Ad_{G} H)-invariant Minkowski form on 𝔤/𝔥. Thus, although Proposition 2.4′ and Theorem 3.5′ are geometric in nature, they can be restated in more algebraic terms. It is in such a form that they are proved in §2 and §3.

Proposition 2.4′ and Theorem 3.5′ are used in work of S. Adams [3] on nontame actions on Lorentz manifolds. See [16, 11, 4, 15, 1, 2] for some other research concerning actions of Lie groups on Lorentz manifolds.

Acknowledgements. I am grateful to Scot Adams for suggesting this problem and providing historical background. I would like to thank the Isaac Newton Institute for Mathematical Sciences for providing the stimulating environment where this work was carried out, and I would also like to thank an anonymous referee for pointing out a misleading passage in the original manuscript. The research was partially supported by a grant from the National Science Foundation (DMS-9801136).

2. Homogeneous spaces of SO(1, n)

The following lemma is elementary.

Lemma 2.1. Let π be the standard representation of 𝔤 = 𝔰𝔬(1, k) on ℝ^{k+1}, and let 𝔤 = 𝔨 + 𝔞 + 𝔫 be an Iwasawa decomposition of 𝔤.

1) The representation π has only one positive weight (with respect to 𝔞), and the corresponding weight space is 1-dimensional.

2) There are subspaces V and W of ℝ^{k+1}, such that

(a) dim(ℝ^{k+1}/V) = 1;

(b) dim W = 1;

(c) π(𝔫)V ⊂ W;

(d) for all nonzero u ∈ 𝔫, we have π(u)^{2}ℝ^{k+1} = W; and

(e) for all nonzero u ∈ 𝔫 and v ∈ ℝ^{k+1}, we have π(u)^{2}v = 0 if and only if v ∈ V.

Corollary 2.2. Let 𝔥 be a subalgebra of a real Lie algebra 𝔤, let Q be a Minkowski form on 𝔤/𝔥, and define π: N_{G}(𝔥) → GL(𝔤/𝔥) by π(g)(v + 𝔥) = (Ad_{G} g)v + 𝔥.

1) Suppose T is a connected Lie subgroup of G that normalizes H, such that π(T) ⊂ SO(Q) and Ad_{G} T is diagonalizable over ℝ. Then, for any ordering of the T -weights on 𝔤, the subalgebra 𝔥 contains codimension-one subspaces of both 𝔤^{+} and 𝔤^{−}, where 𝔤^{+} is the sum of all the positive weight spaces of T, and 𝔤^{−} is the sum of all the negative weight spaces of T.

2) If U is a connected Lie subgroup of G that normalizes H, such that π(U) ⊂ SO(Q) and Ad_{G} U is unipotent, then there are subspaces V/𝔥 and W/𝔥 of 𝔤/𝔥, such that

(a) dim(𝔤/V) = 1;

(b) dim(W/𝔥) = 1;

(c) [V, 𝔲] ⊂ W;

(d) for each u ∈ 𝔲, either W = 𝔥 + (ad_{𝔤} u)^{2}𝔤, or [𝔤, u] ⊂ 𝔥; and
(e) for all u ∈ 𝔲, we have (ad_{𝔤} u)^{2}V ⊂ 𝔥.

For ease of reference, let us record the following well known fact from the theory of real algebraic groups.

Lemma 2.3. Let \overline{H} be a Zariski closed, noncompact subgroup of GL(m, ℝ), for some m. If \overline{H} does not contain any nontrivial hyperbolic elements, then there exist a compact subgroup M and a nontrivial unipotent subgroup U, such that \overline{H} = M ⋉ U.

Proof. The algebraic Levi decomposition [13, Thm. 6.4, p. 286], [7, Prop. 8.4.2, p. 117] provides Zariski closed subgroups M and U of \overline{H}, such that

• \overline{H} = M ⋉ U;

• M is reductive; and

• U is unipotent. Because M is reductive and, being a subgroup of \overline{H}, does not contain hyperbolic elements, we know that M is compact [5, Cor. 9.4, p. 127]. However, M ⋉ U = \overline{H} is not compact, so this implies that U cannot be compact; hence, U is nontrivial. □

Proposition 2.4. Let H be a Lie subgroup of G = SO(1, n), with n ≥ 2, such that • the closure of H is not compact; and

• there is an (Ad_{G} H)-invariant Minkowski form on 𝔤/𝔥. Then either

1) H^{∘} is conjugate to a standard copy of SO(1, n − 1)^{∘} in SO(1, n), or

2) n = 2 and H^{∘} is trivial.

Proof. Let \overline{H} be the Zariski closure of H, and note that the Minkowski form is also invariant under Ad_{G} \overline{H}. Replacing H by a finite-index subgroup, we may assume \overline{H} is Zariski connected.

Let G = KAN be an Iwasawa decomposition of G.

Case 1. Assume n ≥ 3 and A ⊂ \overline{H}. From Corollary 2.2(1) , we see that 𝔥 contains codimension-one subspaces of both 𝔫 and 𝔫^{−}. (Note that this implies H^{∘} is nontrivial.) This implies that \overline{H} is reductive. (Because (H ∩ N)^{∘} unip \overline{H} is a unipotent subgroup that intersects N nontrivially (and ℝ-rank G = 1), it must be contained in N, so unip \overline{H} ⊂ N. Similarly, unip \overline{H} ⊂ N^{−}. Therefore unip \overline{H} ⊂ N ∩ N^{−} = e.) Then, since \overline{H} contains a codimension-one subgroup of N, and since A ⊂ \overline{H}, it follows that \overline{H} is conjugate to either SO(1, n − 1) or SO(1, n). Because H^{∘} is a nontrivial, connected, normal subgroup of \overline{H}, we conclude that H^{∘} is conjugate to either SO(1, n − 1)^{∘} or SO(1, n)^{∘}. Because 𝔤/𝔥 ≠ 0 (else dim 𝔤/𝔥 = 0 < 2, which contradicts the fact that there is a Minkowski form on 𝔤/𝔥), we see that H^{∘} is conjugate to SO(1, n − 1)^{∘}.

Case 2. Assume n ≥ 3 and \overline{H} does not contain any nontrivial hyperbolic elements. From Lemma 2.3, we know there exist a compact subgroup M and a nontrivial unipotent subgroup U, such that \overline{H} = M ⋉ U. Replacing H by a conjugate, we may assume, without loss of generality, that U ⊂ N.

Let us show, for every nonzero u ∈ 𝔲, that [𝔤, u] ⊄ 𝔥. From the Morosov Lemma [8, Thm. 17(1), p. 100], we know there exists v ∈ 𝔤, such that [v, u] is hyperbolic (and nonzero). If [v, u] ∈ 𝔥, this contradicts the fact that \overline{H} does not contain nontrivial hyperbolic elements.

Let V/𝔥 and W/𝔥 be subspaces of 𝔤/𝔥 as in Corollary 2.2(2). Because (ad_{𝔤} u)^{2}𝔤 = 𝔫 for every nonzero u ∈ 𝔫, we have W = 𝔫 + 𝔥 (see 2.2(2d)), so dim 𝔫/(𝔥 ∩ 𝔫) = 1 (see 2.2(2b)) and

[𝔲, V] ⊂ W = 𝔫 + 𝔥 ⊂ 𝔫 + \overline{𝔥} = 𝔫 + 𝔪  (2.5)

(see 2.2(2c)).

Assume, for the moment, that n ≥ 4. Then

dim 𝔲 + dim(V ∩ 𝔫^{−}) ≥ dim(𝔥 ∩ 𝔫) + dim(V ∩ 𝔫^{−})
≥ (dim 𝔫 − 1) + (dim 𝔫^{−} − 1)
= (n − 2) + (n − 2)
≥ n
> dim 𝔫.

This implies that there exist u ∈ 𝔲 and v ∈ V ∩ 𝔫^{−}, such that ⟨u, v⟩ ≅ 𝔰𝔩(2, ℝ), with [u, v] hyperbolic (and nonzero). This contradicts the fact that 𝔪 + 𝔫 has no nontrivial hyperbolic elements.

We may now assume that n = 3. For any nonzero u ∈ 𝔫, we have

dim[u, V] ≥ dim[u, 𝔤] − 1 = dim 𝔫 + 1 > dim 𝔫,

so [𝔲, V] ⊄ 𝔫. Then, from (2.5), we conclude that 𝔪 ≠ 0, so 𝔪 acts irreducibly on 𝔫. This contradicts the fact that 𝔥 ∩ 𝔫 is a codimension-one subspace of 𝔫 that is normalized by 𝔪.

Case 3. Assume n = 2. We may assume H^{∘} is nontrivial (otherwise Conclusion (2) holds). We must have dim 𝔤/𝔥 ≥ 2, so we conclude that dim H^{∘} = 1 and dim 𝔤/𝔥 = 2. Because SO(1, 1) consists of hyperbolic elements, this implies that Ad_{G} h acts diagonalizably on 𝔤/𝔥, for every h ∈ H. Therefore H^{∘} is conjugate to A, and, hence, to SO(1, 1)^{∘}. □

3. Homogeneous spaces of SO(2, n)

Theorem 3.1. (Borel-Tits [6, Prop. 3.1]) Let H be an F -subgroup of a reductive algebraic group G over a field F of characteristic zero. Then there is a parabolic F -subgroup P of G, such that unip H ⊂ unip P and H ⊂ N_{G}(unip H) ⊂ P.

Notation 3.2. Let k = ⌊n/2⌋. Identifying ℂ^{k+1} with ℝ^{2k+2} yields an embedding of SU(1, k) in SO(2, 2k). Then the inclusion ℝ^{2k+2} ↪ ℝ^{2k+3} yields an embedding of SU(1, k) in SO(2, 2k + 1). Thus, we may identify SU(1, ⌊n/2⌋) with a subgroup of SO(2, n).

We use the following well-known result to shorten one case of the proof of Theorem 3.5.

Lemma 3.3. ([12, Lem. 6.8]) If L is a connected, almost-simple subgroup of SO(2, n), such that ℝ-rank L = 1 and dim L > 3, then L is conjugate under O(2, n) to a subgroup of either SO(1, n) or SU(1, ⌊n/2⌋).

Corollary 3.4. Let L be a connected, reductive subgroup of G = SO(2, n), such that ℝ-rank L = 1. Then dim U ≤ n − 1, for every connected, unipotent subgroup U of L.

Furthermore, if dim U = n − 1, then either

1) L is conjugate to SO(1, n)^{∘}; or

2) n is even, and L is conjugate under O(2, n) to SU(1, n/2).

Theorem 3.5. Let H be a Lie subgroup of G = SO(2, n), with n ≥ 3, such that • the closure of H is not compact, and

• there is an (Ad_{G} H)-invariant Minkowski form on 𝔤/𝔥. Then H^{∘} is conjugate to a standard copy of SO(1, n)^{∘} in SO(2, n).

Proof. Let \overline{H} be the Zariski closure of H, and note that the Minkowski form is also invariant under Ad_{G} \overline{H}. Replacing H by a finite-index subgroup, we may assume \overline{H} is Zariski connected.

Let G = KAN be an Iwasawa decomposition of G. For each real root ϕ of 𝔤 (with respect to the Cartan subalgebra 𝔞), let 𝔤_{ϕ} be the corresponding root space, and let proj_{ϕ}: 𝔤 → 𝔤_{ϕ} and proj_{ϕ⊕−ϕ}: 𝔤 → 𝔤_{ϕ} + 𝔤_{−ϕ} be the natural projections. Fix a choice of simple real roots α and β of 𝔤, such that dim 𝔤_{α} = 1 and dim 𝔤_{β} = n − 2 (so the positive real roots are α, β, α + β, and α + 2β). Replacing N by a conjugate under the Weyl group, we may assume 𝔫 = 𝔤_{α} + 𝔤_{β} + 𝔤_{α+β} + 𝔤_{α+2β}. From the classification of parabolic subgroups [5, Prop. 5.14, p. 99], we know that the only proper parabolic subalgebras of 𝔤 that contain 𝔫_{𝔤}(𝔫) are

𝔫_{𝔤}(𝔫), 𝔭_{α} = 𝔫_{𝔤}(𝔫) + 𝔤_{−α}, and 𝔭_{β} = 𝔫_{𝔤}(𝔫) + 𝔤_{−β}.  (3.6)

Case 1. Assume \overline{𝔥} contains nontrivial hyperbolic elements. Let 𝔱 = \overline{𝔥} ∩ 𝔞. Replacing H by a conjugate, we may assume 𝔱 ≠ 0.

Subcase 1.1. Assume 𝔱 ∈ {ker(α + β), ker β}.

Subsubcase 1.1.1. Assume \overline{H} is reductive. We may assume 𝔱 = ker(α+β) (if necessary, replace H with its conjugate under the Weyl reflection corresponding to the root α). Then, from Corollary 2.2(1), we see that 𝔥 contains a codimension-one subspace of 𝔤_{α+2β} + 𝔤_{β} + 𝔤_{−α}. (Note that this implies H^{∘} is nontrivial.)

Let 𝔫′ = 𝔤_{α+β} +𝔤_{α+2β} +𝔤_{β} +𝔤_{−α}, so 𝔫′ is the Lie algebra of a maximal unipotent subgroup of G. (In fact, 𝔫′ is the image of 𝔫 under the Weyl reflection corresponding to the root α.) From the preceding paragraph, we know that

dim(\overline{𝔥} ∩ 𝔫′) ≥ dim(𝔤_{α+2β} + 𝔤_{β} + 𝔤_{−α}) − 1 = n − 1.

Therefore, Corollary 3.4 implies that \overline{H} is conjugate (under O(2, n)) to either SO(1, n) or SU(1, n/2). It is easy to see that \overline{H} is not conjugate to SU(1, n/2). (See [12, proof of Thm. 1.5] for an explicit description of 𝔰𝔲(1, n/2) ∩ 𝔫. If n is even, then n > 3, so 𝔰𝔲(1, n/2) does not contain a codimension-one subspace of any (n − 2)-dimensional root space, but \overline{𝔥} does contain a codimension-one subspace of 𝔤_{β}.) Therefore, we conclude that \overline{H} is conjugate to SO(1, n). Then, because H^{∘} is a nontrivial, connected, normal subgroup of \overline{H}, we conclude that H^{∘} = (\overline{H})^{∘} is conjugate to SO(1, n)^{∘}.

Subsubcase 1.1.2. Assume \overline{H} is not reductive. Let P be a maximal parabolic subgroup of G that contains \overline{H} (see Theorem 3.1). By replacing P and H with conjugate subgroups, we may assume that P contains the minimal parabolic subgroup N_{G}(N). Therefore, the classification of parabolic subalgebras (3.6) implies that P is either P_{α} or P_{β}.

Subsubsubcase 1.1.2.1. Assume 𝔱 = ker(α + β). From Corollary 2.2(1), we see that 𝔥 (and hence also 𝔭) contains codimension-one subspaces of 𝔤_{α+2β} + 𝔤_{β} + 𝔤_{−α} and 𝔤_{−α−2β} + 𝔤_{−β} + 𝔤_{α}. Because 𝔭_{α} does not contain such a subspace of 𝔤_{−α−2β} + 𝔤_{−β} + 𝔤_{α}, we conclude that P = P_{β}. Furthermore, because the intersection of 𝔭_{β} with each of these subspaces does have codimension one, we conclude that 𝔥 has precisely the same intersection; therefore (𝔤_{α+2β} + 𝔤_{β}) + (𝔤_{−β} + 𝔤_{α}) ⊂ 𝔥. Hence 𝔥 ⊃ [𝔤_{α}, 𝔤_{β}] = 𝔤_{α+β}. We now have

(ad_{𝔤} 𝔤_{α+β})^{2}𝔤 = 𝔤_{α} + 𝔤_{α+β} + 𝔤_{α+2β} ≡ 0 (mod 𝔥),

so Corollary 2.2(2d) implies

𝔥 ⊃ [𝔤, 𝔤_{α+β}] ⊃ [𝔤_{−α−β}, 𝔤_{α+β}] ⊃ ker β.

This contradicts the fact that \overline{𝔥} ∩ 𝔞 = 𝔱 = ker(α + β).

Subsubsubcase 1.1.2.2. Assume 𝔱 = ker β. From Corollary 2.2(1), we see that 𝔥 (and hence also 𝔭) contains a codimension-one subspace of 𝔤_{−α} + 𝔤_{−α−β} + 𝔤_{−α−2β}. Because neither 𝔭_{α} nor 𝔭_{β} contains such a subspace, this is a contradiction.

Subcase 1.2. Assume 𝔱 ∈ {ker α, ker(α + 2β)}. We may assume 𝔱 = ker α (if necessary, replace H with its conjugate under the Weyl reflection corresponding to the root β). From Corollary 2.2(1), we see that 𝔥 contains a codimension-one subspace of 𝔤_{β} + 𝔤_{α+β} + 𝔤_{α+2β}. Because any codimension-one subalgebra of a nilpotent Lie algebra must contain the commutator subalgebra, we conclude that 𝔥 contains 𝔤_{α+2β}. Then we have

(ad_{𝔤} 𝔤_{α+2β})^{2}𝔤 = 𝔤_{α+2β} ≡ 0 (mod 𝔥),

so Corollary 2.2(2d) implies

𝔥 ⊃ [𝔤, 𝔤_{α+2β}] ⊃ 𝔤_{β} + 𝔤_{α+β} + 𝔤_{α+2β}.

Similarly, we also have 𝔥 ⊃ 𝔤_{−β} + 𝔤_{−α−β} + 𝔤_{−α−2β}. It is now easy to show that 𝔥 ⊃ 𝔤_{ϕ} for every real root ϕ, so 𝔥 = 𝔤. This contradicts the fact that 𝔤/𝔥 ≠ 0.

Subcase 1.3. Assume 𝔱 contains a regular element of 𝔞. Replacing H by a conjugate under the Weyl group, we may assume that 𝔫 is the sum of the positive root spaces, with respect to 𝔱. Then, from Corollary 2.2(1), we see that 𝔥 contains codimension-one subspaces of both 𝔫 and 𝔫^{−}. Therefore, 𝔥 contains codimension-one subspaces of 𝔤_{β} + 𝔤_{α+β} + 𝔤_{α+2β} and 𝔤_{−β} + 𝔤_{−α−β} + 𝔤_{−α−2β}, so the argument of Subcase 1.2 applies.

Case 2. Assume that \overline{𝔥} does not contain nontrivial hyperbolic elements. From Lemma 2.3, we know there exist a compact subgroup M and a nontrivial unipotent subgroup U, such that \overline{H} = M ⋉ U. Choose subspaces V/𝔥 and W/𝔥 of 𝔤/𝔥 as in Corollary 2.2(2).

Let P be a proper parabolic subgroup of G, such that U ⊂ unip P and H ⊂ P (see Theorem 3.1). Replacing H and P by conjugates, we may assume, without loss of generality, that P contains the minimal parabolic subgroup N_{G}(N) (so unip P ⊂ N). From the classification of parabolic subalgebras (3.6), we know that there are only three possibilities for P. We consider each of these possibilities separately.

First, though, let us show that

for every nonzero u ∈ 𝔲, we have [𝔤, u] ⊄ 𝔥.  (3.7)

From the Morosov Lemma [8, Thm. 17(1), p. 100], we know there exists v ∈ 𝔤, such that [v, u] is hyperbolic (and nonzero). If [v, u] ∈ 𝔥, this contradicts the fact that \overline{𝔥} does not contain nontrivial hyperbolic elements.

Subcase 2.1. Assume P = N_{G}(N) is a minimal parabolic subgroup of G.

Subsubcase 2.1.1. Assume proj_{β} 𝔲 ≠ 0. Choose u ∈ 𝔲, such that proj_{β} u ≠ 0, and let Z = (ad_{𝔤} u)^{2}𝔤_{−α−2β}. (So dim Z = 1, proj_{−α} Z ≠ 0, and proj_{−α−β} Z = 0.) From Corollary 2.2(2d), we know that Z ⊂ W. Then, because proj_{−α} 𝔥 ⊂ proj_{−α} 𝔭 = 0, we conclude, from Corollary 2.2(2b), that W = 𝔥 + Z.

Because W = 𝔥 + Z ⊂ 𝔭 + Z, we have proj_{−α−β} W = 0. Therefore, because proj_{β} u ≠ 0, we conclude, from Corollary 2.2(2c), that proj_{−α−2β} V = 0, so Corollary 2.2(2a) implies that V = ker(proj_{−α−2β}). In particular, we have 𝔤_{−β} ⊂ V, so Corollary 2.2(2c) implies [𝔤_{−β}, u] ⊂ W. Therefore, we have

[𝔤_{−β}, proj_{β} u] ⊂ [𝔤_{−β}, u + (𝔤_{α} + 𝔤_{α+β} + 𝔤_{α+2β})]
= [𝔤_{−β}, u] + [𝔤_{−β}, 𝔤_{α} + 𝔤_{α+β} + 𝔤_{α+2β}]
⊂ W + (𝔤_{α} + 𝔤_{α+β})
= 𝔥 + Z + (𝔤_{α} + 𝔤_{α+β})
⊂ 𝔪 + 𝔫 + Z.

Because proj_{−α}[𝔤_{−β}, proj_{β} u] = 0, we conclude that [𝔤_{−β}, proj_{β} u] ⊂ 𝔪 + 𝔫. This contradicts the fact that 𝔪 + 𝔫 does not contain nontrivial hyperbolic elements.

Subsubcase 2.1.2. Assume proj_{β} 𝔲 = 0. Replacing H by a conjugate under N, we may assume 𝔪 ⊂ 𝔤_{0}, so proj_{β} \overline{𝔥} = 0.

We have 𝔲 ⊂ 𝔤_{α} + 𝔤_{α+β} + 𝔤_{α+2β}, so (ad_{𝔤} u)^{2}𝔤 ⊂ 𝔤_{α} + 𝔤_{α+β} + 𝔤_{α+2β} for every u ∈ 𝔲. Thus, Corollary 2.2(2d) implies W ⊂ (𝔤_{α} + 𝔤_{α+β} + 𝔤_{α+2β}) + 𝔥.

We have

proj_{β⊕−β} W ⊂ proj_{β⊕−β}(𝔤_{α} + 𝔤_{α+β} + 𝔤_{α+2β}) + proj_{β⊕−β} 𝔥 = 0,

so Corollary 2.2(2c) implies that proj_{β⊕−β}((ad_{𝔤} 𝔲)V ) = 0.

Subsubsubcase 2.1.2.1. Assume proj_{α} u ≠ 0, for some u ∈ 𝔲. From the conclusion of the preceding paragraph, we know that proj_{−β}((ad_{𝔤} u)V ) = 0. Because proj_{β} u = 0 and proj_{α} ≠ 0, this implies proj_{−α−β} V = 0, so V = ker(proj_{−α−β}) (see 2.2(2a)). In particular, 𝔤_{−α} ⊂ V, so Corollary 2.2(2c) implies

[𝔤_{α}, 𝔤_{−α}] ⊂ [𝔲 + (𝔤_{α+β} + 𝔤_{α+2β}), 𝔤_{−α}] ⊂ [𝔲, V] + [𝔤_{α+β} + 𝔤_{α+2β}, 𝔤_{−α}]
⊂ W + 𝔤_{β} ⊂ 𝔥 + 𝔫 ⊂ 𝔪 + 𝔫.

This contradicts the fact that 𝔪 + 𝔫 does not contain nontrivial hyperbolic elements.

Subsubsubcase 2.1.2.2. Assume proj_{α+β} u ≠ 0, for some u ∈ 𝔲. From Subsubsubcase 2.1.2.1, we may assume proj_{α} u = 0. Because 0 = proj_{β⊕−β}((ad_{𝔤} u)V ) has codimension ≤ 1 in proj_{β⊕−β}((ad_{𝔤} u)𝔤) (see 2.2(2a)), which contains the 2-dimensional subspace proj_{β⊕−β}([u, 𝔤_{−α−2β} + 𝔤_{−α}]), we have a contradiction.

Subsubsubcase 2.1.2.3. Assume 𝔲 = 𝔤_{α+2β}. (This argument is similar to Subsubsubcase 2.1.2.1.) Because proj_{β}((ad_{𝔤} 𝔲)V ) = 0, we know that proj_{−α−β} V = 0, so V = ker(proj_{−α−β}) (see 2.2(2a)). In particular, 𝔤_{−α−2β} ⊂ V, so Corollary 2.2(2c) implies

[𝔤_{α+2β}, 𝔤_{−α−2β}] ⊂ [𝔲, V] ⊂ W ⊂ 𝔥 + 𝔫 ⊂ 𝔪 + 𝔫.

This contradicts the fact that 𝔪 + 𝔫 does not contain nontrivial hyperbolic elements.

Subcase 2.2. Assume P = P_{α}. We may assume there exists x ∈ 𝔥, such that proj_{−α} x ≠ 0 (otherwise, H ⊂ N_{G}(N), so Subcase 2.1 applies). Note that, because U ⊂ unip P, we have proj_{α} 𝔲 = 0.

Subsubcase 2.2.1. Assume proj_{α+β} 𝔲 ≠ 0. Choose u ∈ 𝔲, such that proj_{α+β} u ≠ 0. Then [x, u] ∈ [𝔥, 𝔲] ⊂ 𝔲, and [[x, u], u] is a nonzero element of 𝔤_{α+2β}, so we see that 𝔤_{α+2β} ⊂ [𝔲, 𝔲]. Because every unipotent subgroup of SO(1, k) is abelian, we conclude that ad_{𝔤} 𝔤_{α+2β} acts trivially on 𝔤/𝔥, which means 𝔥 ⊃ [𝔤, 𝔤_{α+2β}]. This contradicts (3.7).

Subsubcase 2.2.2. Assume proj_{α+β} 𝔲 = 0. We may assume, furthermore, that proj_{α} 𝔥 ≠ 0 (otherwise, by replacing H with its conjugate under the Weyl reflection corresponding to the root α, we could revert to Subcase 2.1). Then, because [𝔥, 𝔲] ⊂ 𝔲, we must have proj_{β} 𝔲 = 0. Thus, 𝔲 = 𝔤_{α+2β}. From Corollary 2.2(2d), we have

W = [𝔤, 𝔤_{α+2β}, 𝔤_{α+2β}] + 𝔥 = 𝔤_{α+2β} + 𝔥 ⊂ 𝔲 + \overline{𝔥} = \overline{𝔥},

so

W ∩ (𝔤_{β} + 𝔤_{α+β}) ⊂ \overline{𝔥} ∩ (𝔤_{β} + 𝔤_{α+β}) = (\overline{𝔥} ∩ 𝔫) ∩ (𝔤_{β} + 𝔤_{α+β})
= 𝔲 ∩ (𝔤_{β} + 𝔤_{α+β}) = 𝔤_{α+2β} ∩ (𝔤_{β} + 𝔤_{α+β}) = 0.

On the other hand, from Corollary 2.2(2c), we know that W contains a codimension-one subspace of [𝔤, 𝔤_{α+2β}], so W contains a codimension-one subspace of 𝔤_{β} + 𝔤_{α+β}. This is a contradiction.

Subcase 2.3. Assume P = P_{β}. Note that, because U ⊂ unip P, we have proj_{β} 𝔲 = 0. From Corollary 2.2(2d), we have

W = 𝔥 + (ad_{𝔤} u)^{2}𝔤 ⊂ 𝔥 + (𝔤_{α} + 𝔤_{α+β} + 𝔤_{α+2β})
= 𝔥 + unip 𝔭_{β} ⊂ (𝔪 + 𝔲) + unip 𝔭_{β} = 𝔪 + unip 𝔭_{β}.

Subsubcase 2.3.1. Assume there is some nonzero u ∈ 𝔲, such that proj_{α} u = 0. Replacing H by a conjugate (under G_{−β}), we may assume proj_{α+β} u ≠ 0.

Let V′ = V ∩ (𝔤_{−α} + 𝔤_{−α−β}). Because V′ contains a codimension-one subspace of 𝔤_{−α} + 𝔤_{−α−β} (see Corollary 2.2(2a)), one of the following two subsubsubcases must apply.

Subsubsubcase 2.3.1.1. Assume there exists v ∈ V′, such that proj_{−α−β} v = 0. From Corollary 2.2(2c), we have [u, v] ∈ W. Then, because [u, v] is a nonzero element of 𝔤_{β}, we conclude that

0 ≠ W ∩ 𝔤_{β} ⊂ (𝔪 + unip 𝔭_{β}) ∩ 𝔤_{β} = 0.

This contradicts the fact that M, being compact, has no nontrivial unipotent elements.

Subsubsubcase 2.3.1.2. Assume proj_{−α−β} V′ = 𝔤_{−α−β}. For v ∈ V′, we have proj_{0}[u, v] = [proj_{α+β} u, proj_{−α−β} v]. Thus, there is some v ∈ V′, such that proj_{0}[u, v] is hyperbolic (and nonzero). On the other hand, from Corollary 2.2(2c), we have [u, v] ∈ W = 𝔪 + unip 𝔭_{β}. This contradicts the fact that 𝔪 ⊂ \overline{𝔥} does not contain nonzero hyperbolic elements.

Subsubcase 2.3.2. Assume proj_{α} u ≠ 0, for every nonzero u ∈ 𝔲. Fix some nonzero u ∈ 𝔲. Because dim 𝔲_{α} = 1, we must have dim 𝔲 = 1 (so 𝔲 = ℝu). Replacing H by a conjugate (under G_{β}), we may assume proj_{α+β} u = 0. Also, we may assume proj_{α+2β} u ≠ 0 (otherwise, we could revert to Subsubcase 2.3.1 by replacing H with its conjugate under the Weyl reflection corresponding to the root β).

Let 𝔱 = [u, 𝔤_{−α} + 𝔤_{−α−2β}]. Because ⟨𝔤_{α}, 𝔤_{−α}⟩ and ⟨𝔤_{α+2β}, 𝔤_{−α−2β}⟩ centralize each other, we see that 𝔱 = [𝔤_{α}, 𝔤_{−α}] + [𝔤_{α+2β}, 𝔤_{−α−2β}] is a two-dimensional subspace of 𝔤 consisting entirely of hyperbolic elements. Because V contains a codimension-one subspace of 𝔤_{−α} + 𝔤_{−α−2β} (see Corollary 2.2(2a)), and [u, V] ⊂ W (see Corollary 2.2(2c)), we see that W contains a codimension-one subspace of 𝔱, so W contains nontrivial hyperbolic elements. This contradicts the fact that W ⊂ 𝔪 + unip 𝔭_{β} does not contain nontrivial hyperbolic elements. □

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Received August 1, 2000